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Probability for IBPS PO

Dice sums, balls drawn from a bag, cards, "at least one" and two people solving the same problem. IBPS PO probability uses a handful of set-ups. The basic formula, counting with combinations, the complement trick, the addition and multiplication rules, seven worked examples and eight practice questions.

11 Oct 2026 6 min read

In this guide
  1. The rules
  2. Know the sample spaces
  3. Drawing from a bag: use combinations
  4. Worked examples
  5. Common mistakes
  6. Practice set
  7. What to do next

Probability questions in IBPS PO are counting questions with a fraction at the end. If you can count the favourable cases and the total cases correctly, the answer follows. Almost every error comes from miscounting, not from the formula.

The topic appears as standalone questions in the prelims, often in a set with a bag of coloured balls, and in quantity comparison. Mains questions add a condition or two but stay within the same set-ups. Master the patterns below and you will recognise most of what is asked.

The rules

  • Probability = favourable outcomes ÷ total outcomes, when all outcomes are equally likely. It always lies between 0 and 1.
  • Complement: P(not A) = 1 − P(A).
  • OR (addition rule): P(A or B) = P(A) + P(B) − P(A and B). If A and B cannot happen together, the last term is zero.
  • AND (multiplication rule): for independent events, P(A and B) = P(A) × P(B).

Why the complement is so useful: "at least one" covers many cases (one, two, three…). Its opposite, "none", is a single case. Counting one case and subtracting from 1 is faster and safer.

Why we subtract the overlap in the OR rule: if you count hearts and then count face cards, the three heart face cards get counted twice. Subtracting them once corrects the double count.

Know the sample spaces

ExperimentTotal outcomesWorth remembering
One coin2
Two coins4HH, HT, TH, TT
Three coins8Exactly two heads: 3 ways
One die6
Two dice36Sum 7 has the most ways (6)
A pack of cards524 suits of 13; 26 red and 26 black; 12 face cards (J, Q, K); 4 of each rank
Choosing r balls from nnCrOrder does not matter

Ways to get each sum with two dice:

Sum23456789101112
Ways12345654321

The pattern rises by one up to 7 and falls by one after it. You never need to list the pairs again.

Drawing from a bag: use combinations

When several balls are drawn together (or one after another without replacement), count with nCr for both the favourable and the total. For combinations, nC2 = n(n − 1)/2 and nC3 = n(n − 1)(n − 2)/6.

Worked examples

The first four examples use one bag: 5 red, 4 blue and 3 green balls (12 in all).

Example 1 (both the same colour): Two balls are drawn at random. Find the probability that both are the same colour.

  • Total = 12C2 = 66.
  • Both red 5C2 = 10; both blue 4C2 = 6; both green 3C2 = 3. Favourable = 19.
  • Probability = 19/66.

Example 2 (one of each): Three balls are drawn. Find the probability that they are all of different colours.

  • Total = 12C3 = 220. Favourable = 5 × 4 × 3 = 60.
  • Probability = 60/220 = 3/11.

Example 3 (at least one): Two balls are drawn. Find the probability that at least one is red.

  • P(no red) = 7C2 ÷ 66 = 21/66.
  • P(at least one red) = 1 − 21/66 = 45/66 = 15/22.

Example 4 (sequential, without replacement): Two balls are drawn one after another without replacement. Find the probability that the first is red and the second is blue.

  • 5/12 × 4/11 = 20/132 = 5/33.
  • Note the order is fixed here. "One red and one blue in any order" would be double this: 10/33, which matches 5 × 4 ÷ 66.

Example 5 (dice): Two dice are thrown. Find the probability that the sum is a prime number.

  • Prime sums: 2, 3, 5, 7, 11. Ways: 1 + 2 + 4 + 6 + 2 = 15.
  • Probability = 15/36 = 5/12.

Example 6 (OR rule with cards): One card is drawn from a pack. Find the probability that it is a heart or a face card.

  • Hearts 13, face cards 12, heart face cards 3.
  • Favourable = 13 + 12 − 3 = 22. Probability = 22/52 = 11/26.

Example 7 (independent events): The probability that A solves a problem is 1/3, and that B solves it is 1/4. Both try independently. Find the probability that the problem is solved.

  • P(neither solves) = 2/3 × 3/4 = 1/2.
  • P(solved) = 1 − 1/2 = 1/2.

Common mistakes

  • Forgetting that "without replacement" reduces the total. The second draw is out of 11, not 12.
  • Missing that order matters in "first red, then blue" but not in "one red and one blue".
  • Double-counting overlaps in OR questions, such as the king of hearts in "a king or a heart".
  • Doing "at least one" the long way. Use 1 − P(none).
  • Adding probabilities of events that can happen together without subtracting the overlap.
  • Multiplying probabilities of events that are not independent, such as two draws without replacement, without adjusting the second fraction.

Practice set

  1. A die is thrown. Find the probability of a number greater than 4.
  2. Two dice are thrown. Find the probability that the sum is 8.
  3. Two dice are thrown. Find the probability of a doublet (both show the same number).
  4. A bag has 4 white and 6 black balls. Two are drawn at random. Find the probability that both are black.
  5. Three coins are tossed. Find the probability of exactly two heads.
  6. One card is drawn from a pack. Find the probability that it is a king or a red card.
  7. A bag has 6 red and 4 green balls. Three are drawn at random. Find the probability that at least one is green.
  8. The probability that A hits a target is 2/5 and that B hits it is 1/2. Each fires once. Find the probability that exactly one of them hits.

Answers:

  1. 1/3. Outcomes 5 and 6: 2/6.
  2. 5/36. (2,6), (3,5), (4,4), (5,3), (6,2).
  3. 1/6. Six doublets out of 36.
  4. 1/3. 6C2 ÷ 10C2 = 15/45.
  5. 3/8. HHT, HTH, THH.
  6. 7/13. 4 kings + 26 red − 2 red kings = 28; 28/52.
  7. 5/6. P(no green) = 6C3 ÷ 10C3 = 20/120 = 1/6.
  8. 1/2. A hits and B misses: 2/5 × 1/2 = 1/5. A misses and B hits: 3/5 × 1/2 = 3/10. Total 1/5 + 3/10 = 1/2.

What to do next

  • Memorise the two-dice sum table and the card facts. They remove most counting errors.
  • Solve one bag-of-balls set (four or five questions) daily for a week, always writing nC2 and nC3 first.
  • Revise permutation and combination, since every bag question is a combination count underneath.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Institute of Banking Personnel Selection website .

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