Class 10 Mathematics

Chapter 14 – Probability Question Bank

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A. Multiple Choice Questions (MCQs)

1.

The theoretical probability of an event $E$ is given by:

(A) $\dfrac{\text{Total outcomes}}{\text{Favourable outcomes}}$

(B) $\dfrac{\text{Favourable outcomes}}{\text{Total possible outcomes}}$

(C) $\text{Favourable outcomes}+\text{Total outcomes}$

(D) $\text{Total outcomes}-\text{Favourable outcomes}$

Answer: (B)

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2.

The probability of an impossible event is:

(A) $1$

(B) $-1$

(C) $0$

(D) $\dfrac12$

Answer: (C)

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3.

The probability of a sure event is:

(A) $0$

(B) $\dfrac12$

(C) $1$

(D) $2$

Answer: (C)

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4.

Which of the following can be the probability of an event?

(A) $-0.4$

(B) $1.3$

(C) $\dfrac34$

(D) $-2$

Answer: (C)

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5.

A fair die is thrown once. The probability of getting a number greater than $4$ is:

(A) $\dfrac16$

(B) $\dfrac13$

(C) $\dfrac12$

(D) $\dfrac23$

Answer: (B)

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6.

A fair die is thrown once. The probability of getting a number less than or equal to $4$ is:

(A) $\dfrac13$

(B) $\dfrac12$

(C) $\dfrac23$

(D) $\dfrac56$

Answer: (C)

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7.

If$$P(E)=0.27,$$

then $P(\overline E)$ is:

(A) $0.27$

(B) $0.63$

(C) $0.73$

(D) $1.27$

Answer: (C)

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8.

A bag contains $3$ blue, $2$ white and $4$ red marbles. The probability of drawing a white marble is:

(A) $\dfrac29$

(B) $\dfrac13$

(C) $\dfrac49$

(D) $\dfrac27$

Answer: (A)

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9.

One card is drawn from a well-shuffled deck of $52$ cards. The probability of getting an ace is:

(A) $\dfrac1{52}$

(B) $\dfrac4{52}$

(C) $\dfrac1{13}$

(D) Both (B) and (C)

Answer: (D)

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10.

The probability of drawing a card that is not an ace from a standard deck is:

(A) $\dfrac1{13}$

(B) $\dfrac4{13}$

(C) $\dfrac{12}{13}$

(D) $\dfrac4{52}$

Answer: (C)

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11.

Two fair coins are tossed simultaneously. The number of equally likely outcomes is:

(A) $2$

(B) $3$

(C) $4$

(D) $6$

Answer: (C)

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12.

Two fair coins are tossed simultaneously. The probability of getting at least one head is:

(A) $\dfrac14$

(B) $\dfrac12$

(C) $\dfrac34$

(D) $1$

Answer: (C)

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13.

Two fair dice are thrown together. The total number of ordered outcomes is:

(A) $12$

(B) $18$

(C) $36$

(D) $72$

Answer: (C)

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14.

When two dice are thrown, the number of outcomes giving a sum of $8$ is:

(A) $4$

(B) $5$

(C) $6$

(D) $8$

Answer: (B)

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15.

The probability of getting a sum of $13$ when two standard dice are thrown is:

(A) $0$

(B) $\dfrac1{36}$

(C) $\dfrac1{13}$

(D) $1$

Answer: (A)

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16.

If$$P(E)=0.05,$$

then $P(\overline E)$ is:

(A) $0.05$

(B) $0.50$

(C) $0.95$

(D) $1.05$

Answer: (C)

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17.

A class has $25$ girls and $15$ boys. One student is selected randomly. The probability of selecting a girl is:

(A) $\dfrac58$

(B) $\dfrac38$

(C) $\dfrac{25}{15}$

(D) $\dfrac12$

Answer: (A)

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18.

A die is thrown once. Which of the following is an impossible event?

(A) Getting an even number

(B) Getting a number less than $7$

(C) Getting $8$

(D) Getting a prime number

Answer: (C)

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19.

A die is thrown once. Which of the following is a sure event?

(A) Getting $6$

(B) Getting an odd number

(C) Getting a number less than $7$

(D) Getting a number greater than $6$

Answer: (C)

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20.

Which of the following statements is correct?

(A) $P(E)>1$ is possible.

(B) $P(E)<0$ is possible.

(C) $0\leq P(E)\leq1$

(D) Every event has probability $\dfrac12$.

Answer: (C)

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B. Fill in the Blanks

1.

The theoretical probability of an event $E$ is the ratio of the number of ________ outcomes to the total number of possible outcomes.

Answer: favourable

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2.

The probability of an impossible event is ________.

Answer: $0$

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3.

The probability of a sure event is ________.

Answer: $1$

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4.

For every event $E$,$$0\leq P(E)\leq\underline{\hspace{1.5cm}}$$

Answer: $1$

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5.

The event representing “not $E$” is called the ________ of $E$.

Answer: complement

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6.

$$P(E)+P(\overline E)=\underline{\hspace{1.5cm}}$$

Answer: $1$

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7.

An event having only one outcome is called an ________ event.

Answer: elementary

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8.

A fair coin has ________ equally likely outcomes.

Answer: $2$

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9.

A standard die has ________ possible outcomes.

Answer: $6$

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10.

When two dice are thrown together, there are ________ ordered outcomes.

Answer: $36$

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11.

A standard deck contains ________ cards.

Answer: $52$

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12.

A standard deck contains ________ aces.

Answer: $4$

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13.

The probability of getting a head when a fair coin is tossed once is ________.

Answer: $\dfrac12$

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14.

The probability of getting a number greater than $4$ on a fair die is ________.

Answer: $\dfrac13$

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15.

The sum of the probabilities of all elementary events of an experiment is ________.

Answer: $1$

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C. True or False

1.

The probability of an event can be greater than $1$.

Answer: False

2.

The probability of an impossible event is $0$.

Answer: True

3.

The probability of a certain event is $1$.

Answer: True

4.

All outcomes of every experiment are necessarily equally likely.

Answer: False

5.

For a fair coin, head and tail are equally likely.

Answer: True

6.

For a standard die, getting $8$ is an impossible event.

Answer: True

7.

For a standard die, getting a number less than $7$ is a sure event.

Answer: True

8.

$$P(E)+P(\overline E)=1$$

Answer: True

9.

When two dice are thrown, there are $11$ equally likely outcomes because the possible sums are $2,3,\ldots,12$.

Answer: False

10.

The ordered pairs $(1,4)$ and $(4,1)$ represent the same outcome.

Answer: False

11.

An elementary event contains exactly one outcome.

Answer: True

12.

A probability of $-0.2$ is possible.

Answer: False

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D. Match the Following

Column A	Column B
1. Impossible event	(a) $1$
2. Sure event	(b) $\dfrac12$
3. Head on a fair coin	(c) $0$
4. Complement of $E$	(d) $1-P(E)$
5. Range of probability	(e) $0\leq P(E)\leq1$

Answers

$$1-(c),\quad 2-(a),\quad 3-(b),\quad 4-(d),\quad 5-(e)$$

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E. Very Short Answer Questions

1.

What are equally likely outcomes?

Answer: Outcomes that have the same chance of occurring.

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2.

What is an elementary event?

Answer: An event containing only one outcome.

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3.

What is an impossible event?

Answer: An event that cannot occur.

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4.

What is a sure event?

Answer: An event that must occur.

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5.

What is the complement of an event $E$?

Answer: The event in which $E$ does not occur.

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6.

Write the range of probability.

Answer:$$0\leq P(E)\leq1$$

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7.

Write the formula for the probability of the complement of $E$.

Answer:$$P(\overline E)=1-P(E)$$

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8.

How many possible outcomes are there when one die is thrown?

Answer: $6$

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9.

How many ordered outcomes are possible when two dice are thrown?

Answer: $36$

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10.

What is the probability of obtaining $8$ on a standard die?

Answer:$$\boxed{0}$$

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F. Short Answer Questions

1.

A fair coin is tossed once. Find the probability of getting a head and the probability of getting a tail.

Solution:

The possible outcomes are$$H,\ T$$

Therefore,$$P(H)=\frac12$$

and$$P(T)=\frac12$$

Answer:$$\boxed{P(H)=P(T)=\frac12}$$

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2.

A die is thrown once. Find the probability of getting:

(i) a number greater than $4$

(ii) a number less than or equal to $4$.

Solution:

Possible outcomes:$$1,2,3,4,5,6$$

For a number greater than $4$, favourable outcomes are$$5,6$$

Hence,$$P(>4)=\frac26=\boxed{\frac13}$$

For a number less than or equal to $4$, favourable outcomes are$$1,2,3,4$$

Hence,$$P(\leq4)=\frac46=\boxed{\frac23}$$

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3.

A bag contains $5$ red balls and $3$ blue balls. One ball is drawn at random. Find the probability that it is:

(i) red

(ii) blue.

Solution:

Total number of balls:$$5+3=8$$

Therefore,$$P(\text{red})=\frac58$$

and$$P(\text{blue})=\frac38$$

Answer:$$\boxed{\frac58,\ \frac38}$$

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4.

If$$P(E)=0.38,$$

find $P(\overline E)$.

Solution:$$P(\overline E)=1-P(E)$$$$=1-0.38$$$$=\boxed{0.62}$$

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5.

A box contains $6$ green and $4$ yellow marbles. Find the probability of not drawing a yellow marble.

Solution:

Not yellow means green.$$P(\text{not yellow}) =\frac6{10}$$$$=\boxed{\frac35}$$

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6.

A standard deck of $52$ cards is well shuffled. Find the probability of drawing:

(i) an ace

(ii) a non-ace.

Solution:

There are $4$ aces.$$P(\text{ace})=\frac4{52} =\boxed{\frac1{13}}$$

There are$$52-4=48$$

non-aces.$$P(\text{non-ace}) =\frac{48}{52} =\boxed{\frac{12}{13}}$$

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7.

A class contains $40$ students, of whom $25$ are girls and $15$ are boys. One student is selected randomly. Find the probability of selecting a boy.

Solution:$$P(\text{boy})=\frac{15}{40}$$$$=\boxed{\frac38}$$

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8.

Two coins are tossed simultaneously. List all possible outcomes.

Answer:$$(H,H),\ (H,T),\ (T,H),\ (T,T)$$

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9.

Two coins are tossed simultaneously. Find the probability of getting exactly one head.

Solution:

Favourable outcomes are$$(H,T),\ (T,H)$$

Therefore,$$P(\text{exactly one head}) =\frac24 =\boxed{\frac12}$$

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10.

Two dice are thrown together. Find the probability that their sum is $8$.

Solution:

Favourable outcomes are$$(2,6),\ (3,5),\ (4,4),\ (5,3),\ (6,2)$$

Thus, there are $5$ favourable outcomes.

Total outcomes:$$6\times6=36$$

Therefore,$$P(\text{sum}=8) =\boxed{\frac5{36}}$$

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G. Application-Based Questions

1. Selection of a Student

A class contains $48$ students. Of these, $28$ are girls and $20$ are boys. One student is selected randomly.

Find:

(i) $P(\text{girl})$

(ii) $P(\text{boy})$

(iii) Verify that the two probabilities add up to $1$.

Solution:$$P(\text{girl})=\frac{28}{48} =\frac7{12}$$$$P(\text{boy})=\frac{20}{48} =\frac5{12}$$

Now,$$\frac7{12}+\frac5{12}=1$$

Hence verified.

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2. Marble Box

A box contains $7$ red, $5$ blue and $8$ green marbles. One marble is selected at random.

Find:

(i) $P(\text{red})$

(ii) $P(\text{blue})$

(iii) $P(\text{green})$

(iv) $P(\text{not green})$

Solution:

Total marbles:$$7+5+8=20$$

Therefore,$$P(\text{red})=\frac7{20}$$$$P(\text{blue})=\frac5{20}=\frac14$$$$P(\text{green})=\frac8{20}=\frac25$$

Using the complement:$$P(\text{not green}) =1-\frac25 =\boxed{\frac35}$$

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3. Two Dice

Two standard dice are thrown together. Find the probability that the sum is:

(i) $5$

(ii) $10$

(iii) $13$

(iv) less than or equal to $12$.

Solution:

Total possible outcomes:$$6\times6=36$$

(i) Sum $=5$

Favourable outcomes:$$(1,4),(2,3),(3,2),(4,1)$$

Hence,$$P(\text{sum}=5) =\frac4{36} =\boxed{\frac19}$$

(ii) Sum $=10$

Favourable outcomes:$$(4,6),(5,5),(6,4)$$

Therefore,$$P(\text{sum}=10) =\frac3{36} =\boxed{\frac1{12}}$$

(iii) Sum $=13$

The greatest possible sum is$$6+6=12$$

Therefore,$$\boxed{P(\text{sum}=13)=0}$$

(iv) Sum $\leq12$

Every possible sum satisfies this condition.

Therefore,$$\boxed{P(\text{sum}\leq12)=1}$$

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4. Selection of a Card

One card is drawn from a standard deck of $52$ cards. Find the probability of getting:

(i) a red card

(ii) a black card

(iii) a king

(iv) a red face card.

Solution:

There are $26$ red cards and $26$ black cards.$$P(\text{red})=\frac{26}{52} =\boxed{\frac12}$$$$P(\text{black})=\frac{26}{52} =\boxed{\frac12}$$

There are $4$ kings.$$P(\text{king})=\frac4{52} =\boxed{\frac1{13}}$$

There are $6$ red face cards.$$P(\text{red face card}) =\frac6{52} =\boxed{\frac3{26}}$$

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H. Assertion–Reason Questions

For each question, choose the correct option:

(A) Both Assertion and Reason are true, and Reason correctly explains Assertion.

(B) Both Assertion and Reason are true, but Reason does not correctly explain Assertion.

(C) Assertion is true, but Reason is false.

(D) Assertion is false, but Reason is true.

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1.

Assertion: The probability of an impossible event is $0$.

Reason: An impossible event has no favourable outcome.

Answer: (A)

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2.

Assertion: The probability of a sure event is $1$.

Reason: Every possible outcome is favourable to a sure event.

Answer: (A)

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3.

Assertion:$$P(\overline E)=1-P(E)$$

Reason: $E$ and $\overline E$ are complementary events.

Answer: (A)

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4.

Assertion: The probability of an event may be $1.4$.

Reason: Probability always lies between $0$ and $1$.

Answer: (D)

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5.

Assertion: When two dice are thrown, there are $36$ equally likely ordered outcomes.

Reason: Each die has $6$ possible outcomes.

Answer: (A)

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6.

Assertion: The sums $2,3,\ldots,12$ are all equally likely when two dice are thrown.

Reason: There are $11$ different possible sums.

Answer: (D)

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I. Case-Based Questions

Case Study 1 — Marbles

A box contains $4$ red, $3$ blue and $5$ green marbles. One marble is selected randomly.

Questions

1. What is the total number of marbles?

(A) $10$

(B) $12$

(C) $14$

(D) $15$

Answer: (B)

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2. Find the probability of selecting a red marble.

Solution:$$P(\text{red})=\frac4{12} =\boxed{\frac13}$$

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3. Find the probability of selecting a blue marble.

Solution:$$P(\text{blue})=\frac3{12} =\boxed{\frac14}$$

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4. Find the probability of not selecting a green marble.

Solution:$$P(\text{not green}) =1-\frac5{12} =\boxed{\frac7{12}}$$

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5. What is the sum of the probabilities of selecting a red, blue or green marble?

Answer:$$\boxed{1}$$

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Case Study 2 — Two Dice

Two standard dice are thrown simultaneously.

Questions

1. How many ordered outcomes are possible?

Answer:$$6\times6=\boxed{36}$$

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2. Write the favourable outcomes for obtaining a sum of $4$.

Answer:$$(1,3),\ (2,2),\ (3,1)$$

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3. Find the probability of obtaining a sum of $4$.

Solution:$$P(\text{sum}=4) =\frac3{36} =\boxed{\frac1{12}}$$

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4. What is the probability of obtaining a sum greater than $12$?

Answer:$$\boxed{0}$$

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5. What is the probability of obtaining a sum less than or equal to $12$?

Answer:$$\boxed{1}$$

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Case Study 3 — Cards

One card is drawn from a well-shuffled standard deck.

Questions

1. How many cards are there in the deck?

Answer:$$\boxed{52}$$

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2. How many aces are there?

Answer:$$\boxed{4}$$

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3. Find the probability of drawing an ace.

Solution:$$P(\text{ace}) =\frac4{52} =\boxed{\frac1{13}}$$

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4. Find the probability of not drawing an ace.

Solution:$$P(\text{not ace}) =1-\frac1{13} =\boxed{\frac{12}{13}}$$

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5. Why can the theoretical probability formula be applied here?

Answer: Because a well-shuffled deck makes each card equally likely to be drawn.

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J. Higher-Order Thinking Questions

1.

A student says:

“When two dice are thrown, the possible sums are $2$ to $12$. Therefore, there are $11$ possible outcomes and each has probability $\dfrac1{11}$.”

Do you agree? Give a reason.

Answer: No. The different sums are not equally likely. Different sums can be obtained in different numbers of ways. Therefore, we must consider the $36$ equally likely ordered outcomes.

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2.

A student says:

“When two coins are tossed, the possible outcomes are HH, TT and HT. Therefore, each outcome has probability $\dfrac13$.”

Is the statement correct? Explain.

Answer: No. The four equally likely outcomes are$$(H,H),\ (H,T),\ (T,H),\ (T,T)$$

The outcomes $HT$ and $TH$ are distinct ordered outcomes.

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3.

The probability of an event $E$ is $0.35$. Find the probability that $E$ does not occur.

Solution:$$P(\overline E)=1-P(E)$$$$=1-0.35$$$$=\boxed{0.65}$$

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4.

A student obtains$$P(E)=\frac98$$

for an event. Is this answer possible? Give a reason.

Answer: No. Since$$0\leq P(E)\leq1,$$

$\frac98>1$, so it cannot be a probability.

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5.

A die is thrown once. A student says that the probability of getting an even number is $\dfrac12$. Is the answer correct? Explain.

Answer: Yes. The favourable outcomes are$$2,4,6$$

out of $6$ possible outcomes.

Therefore,$$P(\text{even})=\frac36=\boxed{\frac12}$$

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K. Mixed Revision Questions

1.

Find the probability of getting a tail when a fair coin is tossed once.$$\boxed{\frac12}$$

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2.

Find the probability of getting a number less than $7$ when a standard die is thrown.$$\boxed{1}$$

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3.

Find the probability of getting $9$ when a standard die is thrown.$$\boxed{0}$$

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4.

If$$P(E)=0.45,$$

find $P(\overline E)$.$$\boxed{0.55}$$

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5.

A bag contains $3$ red and $7$ blue balls. Find the probability of drawing a red ball.$$\boxed{\frac3{10}}$$

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6.

A box contains $5$ red, $4$ white and $6$ green marbles. Find the probability of drawing a green marble.$$\boxed{\frac6{15}=\frac25}$$

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7.

A standard deck of $52$ cards is used. Find the probability of drawing a king.$$\boxed{\frac4{52}=\frac1{13}}$$

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8.

Two coins are tossed. Find the probability of getting two tails.$$\boxed{\frac14}$$

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9.

Two dice are thrown. Find the probability of getting a sum of $7$.

Favourable outcomes:$$(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$$

Therefore,$$P(\text{sum}=7) =\frac6{36} =\boxed{\frac16}$$

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10.

Two dice are thrown. Find the probability of getting a sum of $2$.

Favourable outcome:$$(1,1)$$

Therefore,$$P(\text{sum}=2) =\boxed{\frac1{36}}$$

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L. Important Exam Practice — Without Solutions

1.

A bag contains $6$ red balls and $4$ blue balls. One ball is drawn at random. Find the probability of drawing:

(i) a red ball

(ii) a blue ball.

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2.

A die is thrown once. Find the probability of getting:

(i) an odd number

(ii) a prime number

(iii) a number greater than $3$

(iv) a number less than $6$.

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3.

If$$P(E)=0.72,$$

find $P(\overline E)$.

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4.

A box contains $5$ red, $8$ white and $4$ green marbles. One marble is selected randomly. Find the probability that it is:

(i) red

(ii) white

(iii) not green.

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5.

One card is drawn from a well-shuffled deck of $52$ cards. Find the probability of getting:

(i) a king

(ii) a face card

(iii) a red face card

(iv) a spade.

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6.

Two coins are tossed simultaneously. Find the probability of getting:

(i) two heads

(ii) two tails

(iii) exactly one head

(iv) at least one head.

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7.

Two dice are thrown simultaneously. Find the probability that the sum is:

(i) $6$

(ii) $8$

(iii) $10$

(iv) $12$

(v) $13$.

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8.

A class has $40$ students, of whom $24$ are girls and $16$ are boys. One student is selected randomly. Find:

(i) $P(\text{girl})$

(ii) $P(\text{boy})$

(iii) Verify that$$P(\text{girl})+P(\text{boy})=1.$$

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9.

A standard die is thrown once. Find the probability of getting a number:

(i) greater than $4$

(ii) less than or equal to $4$.

Also verify that the two probabilities are complementary.

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10.

Two dice are thrown. A student claims that every possible sum from $2$ to $12$ has probability $\dfrac1{11}$. Examine the claim and explain the error.

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Final Formula Box

For quick revision:$$\boxed{ P(E)= \frac{\text{Number of favourable outcomes}} {\text{Number of all possible outcomes}} }$$$$\boxed{0\leq P(E)\leq1}$$$$\boxed{P(\text{impossible event})=0}$$$$\boxed{P(\text{sure event})=1}$$$$\boxed{P(\overline E)=1-P(E)}$$$$\boxed{P(E)+P(\overline E)=1}$$$$\boxed{\text{Sum of probabilities of all elementary events}=1}$$

These are the central results emphasized in the uploaded chapter.

I can next turn this into a proper printable question paper + separate answer key, keeping the same mathematical formatting.

now write notes also in this formate

Yes. I’ll write the Probability chapter notes in the same clean mathematical format—proper equations, fractions, symbols, examples, and copy-paste-friendly headings, rather than computer-style text.

Class 10 Mathematics

Chapter 14 – Probability

Complete Chapter Notes

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1. Introduction to Probability

Probability is a measure of the chance or likelihood of an event occurring.

For example:

• When a coin is tossed, it may show a Head or a Tail.
• When a die is thrown, it may show any number from $1$ to $6$.
• When a card is drawn from a well-shuffled deck, different cards have different chances of being selected.

Probability helps us describe these chances mathematically.

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2. Random Experiment

An experiment in which the outcome cannot be predicted with certainty in advance is called a random experiment.

Examples

1. Tossing a coin.

Possible outcomes:$$H,\ T$$

2. Throwing a die.

Possible outcomes:$$1,2,3,4,5,6$$

3. Drawing one card from a well-shuffled deck.

The particular card drawn cannot be predicted beforehand.

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3. Outcome

An outcome is a possible result of a random experiment.

Example

When a die is thrown, the possible outcomes are:$$1,\ 2,\ 3,\ 4,\ 5,\ 6$$

Each of these is an outcome.

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4. Sample Space

The collection of all possible outcomes of a random experiment is called its sample space.

It is usually denoted by $S$.

Example 1: Tossing a Coin

The sample space is$$S=\{H,T\}$$

Therefore, the number of possible outcomes is$$n(S)=2$$

Example 2: Throwing a Die

$$S=\{1,2,3,4,5,6\}$$

Therefore,$$n(S)=6$$

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5. Equally Likely Outcomes

Outcomes are said to be equally likely if each outcome has the same chance of occurring.

Example

For a fair coin:$$P(H)=P(T)=\frac12$$

Therefore, Head and Tail are equally likely outcomes.

Similarly, for a fair die:$$P(1)=P(2)=\cdots=P(6)=\frac16$$

Thus, all six outcomes are equally likely.

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6. Event

An event is a collection of one or more outcomes of a random experiment.

An event is generally denoted by a capital letter such as $E$, $A$, or $B$.

Example

When a die is thrown, let $E$ be the event of getting an even number.

Then,$$E=\{2,4,6\}$$

The favourable outcomes are:$$2,\ 4,\ 6$$

Therefore,$$n(E)=3$$

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7. Favourable Outcomes

The outcomes which satisfy the condition of a given event are called its favourable outcomes.

Example

A die is thrown once.

Let $E$ be the event of getting a number greater than $4$.

Possible outcomes:$$1,2,3,4,5,6$$

Favourable outcomes:$$5,6$$

Therefore,$$\text{Number of favourable outcomes}=2$$

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8. Theoretical Probability

When all possible outcomes of an experiment are equally likely, the probability of an event $E$ is given by$$\boxed{ P(E)= \frac{\text{Number of favourable outcomes}} {\text{Total number of possible outcomes}} }$$

or$$\boxed{ P(E)=\frac{n(E)}{n(S)} }$$

where:

• $n(E)$ = number of favourable outcomes
• $n(S)$ = total number of possible outcomes.

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9. Important Example – Tossing a Coin

A fair coin is tossed once.

The sample space is$$S=\{H,T\}$$

Therefore,$$n(S)=2$$

Probability of getting Head

There is one favourable outcome.$$n(H)=1$$

Hence,$$P(H)=\frac12$$

Probability of getting Tail

Similarly,$$P(T)=\frac12$$

Therefore,$$P(H)+P(T) = \frac12+\frac12 =1$$

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10. Probability of an Event in a Die Experiment

A fair die is thrown once.

The sample space is$$S=\{1,2,3,4,5,6\}$$

Therefore,$$n(S)=6$$

Example

Find the probability of getting an even number.

The favourable outcomes are$$E=\{2,4,6\}$$

Thus,$$n(E)=3$$

Therefore,$$P(E)=\frac36$$$$\boxed{P(E)=\frac12}$$

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11. Impossible Event

An event which cannot occur is called an impossible event.

The probability of an impossible event is$$\boxed{0}$$

Example

A die is thrown once. Find the probability of getting $8$.

Since a standard die has only the numbers$$1,2,3,4,5,6,$$

getting $8$ is impossible.

Therefore,$$\boxed{P(8)=0}$$

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12. Sure Event

An event which must occur is called a sure event or certain event.

The probability of a sure event is$$\boxed{1}$$

Example

A die is thrown once. Find the probability of getting a number less than $7$.

Every possible outcome is less than $7$:$$1,2,3,4,5,6$$

Therefore,$$P(\text{number}<7) =\frac66$$$$\boxed{P(\text{number}<7)=1}$$

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13. Range of Probability

For every event $E$,$$\boxed{0\leq P(E)\leq1}$$

This means:

• Probability can never be negative.
• Probability can never be greater than $1$.

Important

$$P(E)=0$$

represents an impossible event.$$P(E)=1$$

represents a sure event.

For an event that may or may not occur,$$0<P(E)<1$$

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14. Elementary Event

An event containing only one outcome is called an elementary event.

Example

A die is thrown once.

The event of getting $3$ is$$E=\{3\}$$

It contains only one outcome.

Therefore, $E$ is an elementary event.

Its probability is$$P(E)=\frac16$$

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15. Complementary Events

If $E$ is an event, then the event that $E$ does not occur is called the complement of $E$.

The complement of $E$ is denoted by$$\overline E$$

or sometimes $E’$.

For complementary events,$$\boxed{ P(E)+P(\overline E)=1 }$$

Therefore,$$\boxed{ P(\overline E)=1-P(E) }$$

and$$\boxed{ P(E)=1-P(\overline E) }$$

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16. Example of Complementary Events

If$$P(E)=0.35,$$

find $P(\overline E)$.

Using$$P(\overline E)=1-P(E),$$

we get$$P(\overline E)=1-0.35$$$$\boxed{P(\overline E)=0.65}$$

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17. Probability of “Not” an Event

Many probability questions contain words such as:

• not
• does not occur
• other than
• neither
• without

These often indicate the complement of an event.

Example

A die is thrown once. Find the probability of not getting a 66.

Probability of getting $6$:$$P(6)=\frac16$$

Therefore,$$P(\text{not }6) = 1-\frac16$$$$\boxed{\frac56}$$

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18. Tossing Two Coins

When two coins are tossed simultaneously, the possible ordered outcomes are$$(H,H),\ (H,T),\ (T,H),\ (T,T)$$

Thus,$$n(S)=4$$

All four outcomes are equally likely when the coins are fair.

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Example: Probability of Getting Two Heads

Only one outcome gives two heads:$$(H,H)$$

Therefore,$$P(\text{two heads}) = \frac14$$$$\boxed{\frac14}$$

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Example: Probability of Getting Exactly One Head

The favourable outcomes are$$(H,T),\ (T,H)$$

Therefore,$$P(\text{exactly one head}) = \frac24$$$$\boxed{\frac12}$$

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Example: Probability of Getting At Least One Head

The outcomes containing at least one head are$$(H,H),\ (H,T),\ (T,H)$$

Therefore,$$P(\text{at least one head}) = \frac34$$$$\boxed{\frac34}$$

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19. Throwing Two Dice

When two standard dice are thrown together, each die has $6$ possible outcomes.

Therefore, the total number of ordered outcomes is$$6\times6=36$$

Thus,$$\boxed{n(S)=36}$$

The outcomes can be represented by ordered pairs:$$(1,1),(1,2),\ldots,(6,6)$$

The first number represents the result on the first die and the second number represents the result on the second die.

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20. Important Point About Two Dice

When two dice are thrown, the sums are not equally likely.

For example, a sum of $2$ can occur only in one way:$$(1,1)$$

But a sum of $7$ can occur in six ways:$$(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$$

Therefore,$$P(\text{sum}=2)=\frac1{36}$$

whereas$$P(\text{sum}=7)=\frac6{36} =\frac16$$

Hence, the different sums do not have equal probabilities.

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21. Example – Sum of Two Dice is 8

Two dice are thrown together. Find the probability that the sum is $8$.

The favourable outcomes are$$(2,6),(3,5),(4,4),(5,3),(6,2)$$

Therefore,$$n(E)=5$$

Total outcomes:$$n(S)=36$$

Hence,$$P(E)=\frac5{36}$$$$\boxed{P(\text{sum}=8)=\frac5{36}}$$

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22. Example – Sum of Two Dice is 7

The favourable outcomes are$$(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$$

Thus,$$n(E)=6$$

Therefore,$$P(\text{sum}=7) = \frac6{36}$$$$\boxed{\frac16}$$

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23. Impossible and Sure Events with Two Dice

The smallest possible sum when two dice are thrown is$$1+1=2$$

and the largest possible sum is$$6+6=12$$

Therefore:

Sum greater than $12$

This is impossible.$$\boxed{P(\text{sum}>12)=0}$$

Sum less than or equal to $12$

This is certain.$$\boxed{P(\text{sum}\leq12)=1}$$

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24. Cards

A standard deck contains$$\boxed{52\text{ cards}}$$

The deck contains:

• $4$ suits
• $13$ cards in each suit
• $26$ red cards
• $26$ black cards
• $4$ aces
• $4$ kings
• $4$ queens
• $4$ jacks

The face cards are:$$J,\ Q,\ K$$

There are $12$ face cards in total.

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25. Probability of Drawing an Ace

There are $4$ aces in a deck of $52$ cards.

Therefore,$$P(\text{ace}) = \frac4{52}$$$$\boxed{P(\text{ace})=\frac1{13}}$$

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26. Probability of Not Drawing an Ace

There are $48$ non-ace cards.

Therefore,$$P(\text{not ace}) = \frac{48}{52}$$$$\boxed{P(\text{not ace})=\frac{12}{13}}$$

Alternatively,$$P(\text{not ace}) = 1-\frac1{13} = \frac{12}{13}$$

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27. Probability of Drawing a Red Card

There are $26$ red cards.

Therefore,$$P(\text{red card}) = \frac{26}{52}$$$$\boxed{\frac12}$$

Similarly,$$P(\text{black card}) = \frac{26}{52} = \boxed{\frac12}$$

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28. Experimental Probability

Probability can also be estimated from actual observations.

If an experiment is performed $n$ times and an event $E$ occurs $r$ times, then its experimental probability is$$\boxed{ P(E)=\frac{r}{n} }$$

where:

• $r$ = number of times the event occurs
• $n$ = total number of trials.

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29. Example of Experimental Probability

Suppose a coin is tossed $100$ times and Head occurs $56$ times.

Then the experimental probability of getting Head is$$P(H)=\frac{56}{100}$$$$\boxed{P(H)=0.56}$$

This is an experimental estimate based on the actual trials.

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30. Theoretical Probability vs Experimental Probability

Theoretical Probability

It is calculated using the possible outcomes of an experiment.$$\boxed{ P(E)= \frac{\text{Favourable outcomes}} {\text{Total possible outcomes}} }$$

Experimental Probability

It is calculated from actual observations.$$\boxed{ P(E)= \frac{\text{Number of times the event occurs}} {\text{Number of trials}} }$$

Experimental probability may vary from one set of trials to another.

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31. Important Properties of Probability

For an event $E$:

Property 1

$$\boxed{0\leq P(E)\leq1}$$

Property 2

For an impossible event:$$\boxed{P(E)=0}$$

Property 3

For a sure event:$$\boxed{P(E)=1}$$

Property 4

For the complement of $E$:$$\boxed{P(\overline E)=1-P(E)}$$

Property 5

For complementary events:$$\boxed{P(E)+P(\overline E)=1}$$

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32. How to Solve Probability Questions

Follow these steps:

Step 1: Identify the experiment

For example:

• coin
• die
• cards
• selection of an object

Step 2: Find the total number of possible outcomes

Write the sample space or calculate its size.

Step 3: Identify favourable outcomes

Select the outcomes that satisfy the given condition.

Step 4: Apply the formula

$$P(E)=\frac{n(E)}{n(S)}$$

Step 5: Simplify the answer

Write the probability in its simplest form.

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33. Common Mistakes to Avoid

Mistake 1: Probability greater than $1$

An answer such as$$\frac54$$

cannot be a probability because$$P(E)\leq1$$

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Mistake 2: Negative probability

An answer such as$$-\frac12$$

is impossible because$$P(E)\geq0$$

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Mistake 3: Counting sums instead of outcomes

For two dice, the possible sums are$$2,3,\ldots,12$$

but these $11$ sums are not equally likely.

The equally likely outcomes are the $36$ ordered pairs.

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Mistake 4: Treating $HT$ and $TH$ as one outcome

For two coin tosses,$$(H,T)\neq(T,H)$$

when outcomes are recorded in order.

Therefore, there are four equally likely outcomes:$$(H,H),(H,T),(T,H),(T,T)$$

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34. Quick Revision Table

Concept	Important Result
Probability	$0\leq P(E)\leq1$
Impossible event	$P(E)=0$
Sure event	$P(E)=1$
Theoretical probability	$\displaystyle P(E)=\frac{n(E)}{n(S)}$
Complement	$\displaystyle P(\overline E)=1-P(E)$
Complementary events	$\displaystyle P(E)+P(\overline E)=1$
Fair coin	$P(H)=P(T)=\frac12$
One die	$6$ outcomes
Two dice	$36$ ordered outcomes
Standard deck	$52$ cards
Aces	$4$
Red cards	$26$
Black cards	$26$

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35. Formula Sheet

Theoretical Probability

$$\boxed{ P(E)=\frac{n(E)}{n(S)} }$$

Complement

$$\boxed{ P(\overline E)=1-P(E) }$$

Sum of Complementary Probabilities

$$\boxed{ P(E)+P(\overline E)=1 }$$

Probability Range

$$\boxed{ 0\leq P(E)\leq1 }$$

Impossible Event

$$\boxed{ P(E)=0 }$$

Sure Event

$$\boxed{ P(E)=1 }$$

Experimental Probability

$$\boxed{ P(E)= \frac{\text{Number of times event occurs}} {\text{Total number of trials}} }$$

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36. One-Minute Revision

Remember these six points before the examination:

1. Probability measures the chance of an event occurring.
2. For equally likely outcomes,

$$P(E)=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}$$

3. Probability always lies between $0$ and $1$:

$$0\leq P(E)\leq1$$

4. Impossible event:

$$P(E)=0$$

5. Sure event:

$$P(E)=1$$

6. Complementary event:

P(E)=1−P(E)