Basic Probability – Notes for Competitive Exams
Introduction
Probability is an important topic in the Quantitative Aptitude section of competitive exams like SSC, Banking, Railway, Defence, and State Exams.
It helps measure the likelihood of an event occurring. Probability questions are generally easy to moderate and can save time if practiced well.
Probability Formula
The probability of an event (P) is calculated as:P(E)=Total number of outcomesNumber of favorable outcomes
Where:
- 0 ≤ P(E) ≤ 1
- P(E) = 0 → Event cannot occur
- P(E) = 1 → Event is certain to occur
Types of Probability Questions
1. Single Event Probability
- Probability of a single event happening.
Example: Probability of getting a 4 on a dice = 1/6
2. Multiple Events Probability
- Probability of multiple events occurring together.
Example: Probability of getting two heads in two coin tosses = 1/4
3. Complementary Probability
- Probability that an event does not happen:
P(Not E)=1−P(E)
4. Conditional Probability
- Probability of an event given that another event has already occurred.
P(A∣B)=P(B)P(A∩B)
5. Independent and Dependent Events
- Independent events: Outcome of one does not affect the other
- Dependent events: Outcome of one affects the other
Common Tricks for Probability Questions
- 🔢 Count the total possible outcomes carefully
- 📝 Identify favorable outcomes
- ➕ Use addition rule for mutually exclusive events
- ✖️ Use multiplication rule for independent events
- ⏱ Practice dice, coins, cards, and balls-based questions
Examples
- Coin Toss: Probability of getting a head = 1/2
- Dice Roll: Probability of getting an even number = 3/6 = 1/2
- Card Selection: Probability of getting a King from a deck of 52 cards = 4/52 = 1/13
Basic Probability Questions
Q1. A single die is rolled. What is the probability of getting a 5?
Q2. A coin is tossed. What is the probability of getting a tail?
Q3. Two dice are rolled. What is the probability of getting a double 6?
Q4. A card is drawn from a standard deck of 52 cards. Probability of getting a Queen?
Q5. A bag contains 5 red balls and 7 blue balls. A ball is drawn at random. Probability it is red?
Q6. One die is rolled. Probability of getting an even number?
Q7. A box contains 3 white, 4 black, and 5 red balls. One ball is drawn. Probability it is not black?
Q8. Two coins are tossed together. Probability of getting exactly one head?
Q9. A bag has 6 green, 3 yellow, and 1 blue ball. One ball is picked at random. Probability it is green?
Q10. Probability of getting a number divisible by 3 when a die is rolled?
Q11. A card is drawn from a deck. Probability it is a heart?
Q12. A bag contains 4 red and 6 black balls. Two balls are drawn without replacement. Probability both are red?
Q13. A die is rolled twice. Probability the sum of numbers is 7?
Q14. A bag contains 5 white and 3 black balls. One ball is drawn. Probability it is not white?
Q15. Probability of getting at least one head when two coins are tossed?
Q16. Probability of getting a number less than 4 when a die is rolled?
Q17. A card is drawn. Probability it is either King or Queen?
Q18. Probability of not getting a tail when a coin is tossed?
Q19. Two dice are rolled. Probability the sum is even?
Q20. Probability of drawing an Ace from a standard deck of cards?
Q21. A bag contains 3 red and 2 green balls. Two balls are drawn with replacement. Probability both are red?
Q22. Probability of drawing a black card or a red King from a deck?
Q23. Probability of getting two heads in three tosses of a coin?
Q24. Probability of getting a number greater than 4 when a die is rolled?
Q25. Probability of not drawing a spade from a standard deck of 52 cards?
Answers
Q1. 1/6
Q2. 1/2
Q3. 1/36
Q4. 4/52 = 1/13
Q5. 5/12
Q6. 3/6 = 1/2
Q7. 8/12 = 2/3
Q8. 2/4 = 1/2
Q9. 6/10 = 3/5
Q10. 2/6 = 1/3
Q11. 13/52 = 1/4
Q12. (4/10) × (3/9) = 2/15
Q13. 6/36 = 1/6
Q14. 3/8
Q15. 3/4
Q16. 3/6 = 1/2
Q17. (4+4)/52 = 8/52 = 2/13
Q18. 1 – 1/2 = 1/2
Q19. 18/36 = 1/2
Q20. 4/52 = 1/13
Q21. (3/5) × (3/5) = 9/25
Q22. (26+4)/52 = 30/52 = 15/26
Q23. 3/8
Q24. 2/6 = 1/3
Q25. 39/52 = 3/4