1. Basics
P(E) = favourable outcomes / total outcomes. 0 ≤ P ≤ 1. P(sure event) = 1, P(impossible) = 0. P(not E) = 1 − P(E).
A probability can never be negative or more than 1 (e.g. −1.5 or 1.2 are impossible).
2. Coins
| Coins | Total outcomes |
|---|---|
| 1 | 2 |
| 2 | 4 (HH, HT, TH, TT) |
| 3 | 8 |
| n | 2ⁿ |
- Exactly k heads from n coins: ⁿCₖ/2ⁿ.
- At least one head: 1 − 1/2ⁿ.
(ASO 2021) 4 coins, exactly 2 heads → 6/16 = 3/8.
3. Dice
- One die: 6 outcomes. Two dice: 36.
- Sums with two dice: 2 and 12 → 1 way; 3, 11 → 2; 4, 10 → 3; 5, 9 → 4; 6, 8 → 5; 7 → 6 ways.
- Doublets: 6/36 = 1/6; different numbers: 30/36 = 5/6.
Two dice, sum 8 → 5/36. Sum is a prime (2, 3, 5, 7, 11) → 1 + 2 + 4 + 6 + 2 = 15/36 = 5/12.
4. Cards (52)
- 4 suits × 13: spades ♠ and clubs ♣ black; hearts ♥ and diamonds ♦ red.
- Face cards (J, Q, K): 12; with aces as honours: 16.
- 4 each of every rank (aces, kings, …).
P(king) = 4/52 = 1/13. P(red face card) = 6/52 = 3/26. P(club) = 13/52 = 1/4. P(red king) = 1/26.
5. Calendar and numbers
(ASO 2021) Non-leap year has 53 Sundays: 1 extra day → 1/7. Leap year: 2 extra days → 2/7.
(ASO 2021) Prime from 1–25: 9 primes → 9/25.
(ASO 2012) Two friends, different birthdays → 364/365.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. A fair die is thrown once. The probability of getting greater than 3 is:
(a) 1/2 (b) 1/6 (c) 1/3 (d) 2/3
Answer: (a) 1/2
Solution: Favourable outcomes = 3 out of 6. P = 1/2. (Asked in OPSC ASO 2015, 2021.)
Example 2. A card is drawn from a well-shuffled deck of 52 cards. The probability that it is an ace of spades is:
(a) 1/52 (b) 1/13 (c) 1/26 (d) 2/52
Answer: (a) 1/52
Solution: Favourable = 1. P = 1/52 = 1/52.
Example 3. A die is thrown once. The probability of getting a multiple of 3 is:
(a) 2/3 (b) 1/6 (c) 1/3 (d) 1/2
Answer: (c) 1/3
Solution: Multiples of 3: 3, 6 → 2/6 = 1/3.
ASO exam level
Example 4. 4 coins are tossed together. The probability of getting exactly 4 heads is:
(a) 15/16 (b) 1/16 (c) 1 (d) 1/8
Answer: (b) 1/16
Solution: Total outcomes = 2^4 = 16. Ways to choose 4 heads = 4C4 = 1. P = 1/16.
Example 5. The probability that a non-leap year has 53 Sundays is:
(a) 6/7 (b) 2/7 (c) 5/7 (d) 1/7
Answer: (d) 1/7
Solution: A non-leap year has 52 weeks and 1 extra day. The extra day is Sunday in 1 of 7 cases: P = 1/7.
Example 6. The probability that a leap year has 53 Sundays is:
(a) 2/7 (b) 5/7 (c) 1/7 (d) 6/7
Answer: (a) 2/7
Solution: A leap year has 52 weeks and 2 extra days. 2 of the 7 possible pairs contain Sunday: P = 2/7.
Example 7. In a cricket match, a batsman hits a boundary 9 times out of 36 balls. Find the probability that he does not hit a boundary.
(a) 1/4 (b) 1/2 (c) 3/4 (d) 2/3
Answer: (c) 3/4
Solution: P(boundary) = 1/4. P(not) = 1 − 1/4 = 3/4.
Example 8. A number is selected at random from 1 to 30. The probability that it is prime is:
(a) 1/2 (b) 1/3 (c) 11/30 (d) 3/10
Answer: (b) 1/3
Solution: Count = 10. P = 10/30 = 1/3.
Example 9. A number is selected at random from 1 to 25. The probability that it is prime is:
(a) 8/25 (b) 2/5 (c) 1/2 (d) 9/25
Answer: (d) 9/25
Solution: Count = 9. P = 9/25 = 9/25.
Example 10. Two dice are thrown together. The probability that the sum is 9 is:
(a) 1/9 (b) 1/12 (c) 5/36 (d) 1/6
Answer: (a) 1/9
Solution: Pairs adding to 9: 4 of 36 → 1/9.
Example 11. Two dice are thrown together. The probability that the sum is 3 is:
(a) 1/12 (b) 1/18 (c) 1/6 (d) 1/36
Answer: (b) 1/18
Solution: Pairs adding to 3: 2 of 36 → 1/18.
Example 12. One card is drawn from a well-shuffled deck of 52. The probability that it is a black face card is:
(a) 23/26 (b) 7/52 (c) 3/26 (d) 1/9
Answer: (c) 3/26
Solution: Favourable = 6. P = 6/52 = 3/26.
Example 13. 6 coins are tossed. The probability of getting at least one head is:
(a) 6/64 (b) 1/64 (c) 63/64 (d) 1/2
Answer: (c) 63/64
Solution: P(no head) = 1/64. At least one = 1 − 1/64 = 63/64.
Advanced level
Example 14. Two dice are thrown together. The probability that the sum is 10 is:
(a) 1/6 (b) 1/9 (c) 1/12 (d) 1/4
Answer: (c) 1/12
Solution: There are 36 outcomes; pairs giving 10: 3. P = 1/12.
Example 15. Two dice are thrown together. The probability that the sum is 6 is:
(a) 6/36 (b) 1/6 (c) 5/12 (d) 5/36
Answer: (d) 5/36
Solution: There are 36 outcomes; pairs giving 6: 5. P = 5/36.
Example 16. 5 coins are tossed together. The probability of getting exactly 1 heads is:
(a) 3/16 (b) 1/10 (c) 1/6 (d) 5/32
Answer: (d) 5/32
Solution: Ways = 5C1 = 5 out of 2^5 = 32 → 5/32.
Topic-wise practice for Probability I: coins, dice, cards (Quant): Simple events. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Probability of a simple event; P(E) + P(not E) = 1
- One, two or three coins
- One or two dice: sums and doubles
- A pack of 52 cards
- Numbers 1 to 100: primes, multiples
- Letters of a word, days and leap year