1. Drawing one ball
P = balls of that colour / total. "Not red" = 1 − P(red).
(ASO 2021) 3 green, 4 blue, 2 orange → not orange = 7/9.
2. Drawing two or more together (combinations)
Use ⁿCᵣ = n!/(r!(n − r)!). ⁿC₂ = n(n − 1)/2.
- Both red from r red and b blue: ʳC₂/ⁿC₂.
- One of each: (r × b)/ⁿC₂.
- At least one red: 1 − ᵇC₂/ⁿC₂.
(ASO 2012) 2 boys, 3 girls, choose 2, at least one boy → 1 − ³C₂/⁵C₂ = 1 − 3/10 = 7/10.
Bag: 5 red, 3 black; two drawn → both red = 10/28 = 5/14; one of each = 15/28.
3. With and without replacement
- With replacement: events independent; multiply the same fractions: (r/n)(r/n).
- Without replacement: second fraction changes: (r/n)((r − 1)/(n − 1)).
4. Addition rule
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Mutually exclusive (cannot happen together): P(A ∩ B) = 0.
- Tickets 1–N, multiple of a or b: count multiples of a + of b − of LCM.
Tickets 1–30, multiple of 3 or 5: 10 + 6 − 2 = 14 → 7/15.
5. Independent events
P(A ∩ B) = P(A) × P(B). If A, B independent then A and B′ are independent too.
(ASO 2021) P(A ∪ B′) = 0.8, P(A) = 0.3 → 0.3 + 0.7 P(B′) = 0.8 → P(B′) = 5/7 → P(B) = 2/7.
A hits a target with P 1/2, B with 1/3. P(target hit) = 1 − (1/2)(2/3) = 2/3.
6. Letters in a word
(ASO 2021) 'ASPIRATION': 10 letters, vowels A, I, A, I, O → 1/2.
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 bag has 5 green, 3 blue and 2 orange marbles. One marble is drawn at random. The probability that it is blue is:
(a) 3/7 (b) 1/3 (c) 3/10 (d) 7/10
Answer: (c) 3/10
Solution: P = 3/10 = 3/10.
Example 2. A bag has 9 red, 6 blue and 5 green balls. One ball is drawn at random. The probability that it is not green is:
(a) 1/4 (b) 2/3 (c) 4/5 (d) 3/4
Answer: (d) 3/4
Solution: Not green = 15 of 20 → 3/4.
ASO exam level
Example 3. A letter is chosen at random from the word 'ODISHA'. The probability that it is a vowel is:
(a) 2/3 (b) 5/26 (c) 1/2 (d) 1/3
Answer: (c) 1/2
Solution: Letters = 6, vowels = 3. P = 1/2.
Example 4. A and B are independent events with P(A ∪ B) = 0.8 and P(A) = 0.3. Then P(B) = ?
(a) 2/7 (b) 5/14 (c) 1/2 (d) 5/7
Answer: (d) 5/7
Solution: P(A∪B) = P(A) + P(B) − P(A)P(B) → 0.8 = 0.3 + P(B)(1 − 0.3) → P(B) = 5/7.
Example 5. A box has 3 white and 4 black balls. Two balls are drawn at random. The probability that both are white is:
(a) 1/7 (b) 9/49 (c) 2/7 (d) 3/7
Answer: (a) 1/7
Solution: 3C2 / 7C2 = 3/21 = 1/7.
Example 6. Tickets numbered 1 to 80 are mixed and one is drawn. The probability that its number is a multiple of 7 or 5 is:
(a) 11/80 (b) 5/16 (c) 27/80 (d) 3/10
Answer: (b) 5/16
Solution: Multiples of 7: 11, of 5: 16, of both: 2. Total = 25. P = 5/16.
Example 7. Tickets numbered 1 to 30 are mixed and one is drawn. The probability that its number is a multiple of 2 or 4 is:
(a) 1/2 (b) 1/3 (c) 7/15 (d) 11/15
Answer: (a) 1/2
Solution: Multiples of 2: 15, of 4: 7, of both: 7. Total = 15. P = 1/2.
Advanced level
Example 8. Two children are chosen at random from 4 boys and 2 girls. The probability that at least one is a boy is:
(a) 1/15 (b) 2/3 (c) 8/15 (d) 14/15
Answer: (d) 14/15
Solution: Total = 6C2 = 15. No boy = 2C2 = 1. P(at least one boy) = 1 − 1/15 = 14/15.
Example 9. x is chosen from 1, 2, 3 and y from 1, 4, 9. What is P(xy < 9)?
(a) 5/9 (b) 2/3 (c) 4/9 (d) 1/2
Answer: (a) 5/9
Solution: 9 pairs. xy < 9 for (1,1),(1,4),(2,1),(2,4),(3,1) = 5. P = 5/9.
Example 10. x is chosen from 1, 2, 3 and y from 1, 4, 9. What is P(xy < 9)?
(a) 5/9 (b) 2/3 (c) 1/2 (d) 4/9
Answer: (a) 5/9
Solution: 9 pairs. xy < 9 for (1,1),(1,4),(2,1),(2,4),(3,1) = 5. P = 5/9.
Example 11. From 2 boys and 6 girls, two children are selected at random. The probability that at least one boy is selected is:
(a) 1/4 (b) 1/28 (c) 15/28 (d) 13/28
Answer: (d) 13/28
Solution: 1 − P(both girls) = 1 − 6C2/8C2 = 1 − 15/28 = 13/28.
Topic-wise practice for Probability II: balls, events (Quant): Drawing objects, combined events. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Drawing balls of a colour from a bag
- Drawing two or more objects together
- With and without replacement
- Socks, marbles and tickets
- P(A ∪ B), mutually exclusive and independent events