Class 10 Mathematics Chapter 14 – Probability

---

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.

---

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.

---

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.

---

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,$$n(S)=6$$

---

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.

---

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,$$n(E)=3$$

---

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

---

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.

---

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

---

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

---

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

---

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

---

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,$$0<P(E)<1$$

---

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$$P(E)=\frac16$$

---

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

---

16. Example of Complementary Events

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

find $P(\overline E)$.

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

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

---

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

---

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.

---

Example: Probability of Getting Two Heads

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

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

---

Example: Probability of Getting Exactly One Head

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

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

---

Example: Probability of Getting At Least One Head

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

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

---

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.

---

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.

---

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

---

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

---

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

---

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.

---

25. Probability of Drawing an Ace

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

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

---

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

---

27. Probability of Drawing a Red Card

There are $26$ red cards.

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

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

---

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.

---

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.

---

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.

---

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

---

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.

---

33. Common Mistakes to Avoid

Mistake 1: Probability greater than $1$

An answer such as$$\frac54$$

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

---

Mistake 2: Negative probability

An answer such as$$-\frac12$$

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

---

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.

---

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:$$(H,H),(H,T),(T,H),(T,T)$$

---

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	$26$

---

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

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

---

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)