Probability for IBPS Clerk

Probability questions in the clerk exam are short, and most of them test one skill: counting. If you can list or count every possible outcome and then count the ones the question wants, the answer is a single division. The questions go wrong when the counting goes wrong, so this guide spends most of its time on counting safely.

Probability is not a fixed feature of every paper. It shows up now and then as a standalone question, usually on coins, dice, cards or coloured balls. Treat it as a quick-win topic: a few hours of practice covers almost everything asked at this level.

The formula, and why it works

Probability of an event = number of favourable outcomes ÷ total number of outcomes.

This only works when every outcome is equally likely. A fair die has six faces and each is as likely as any other, so the chance of a 4 is 1/6. That is why you must count outcomes that really are equally likely. For two dice, "sum = 7" and "sum = 2" are not equally likely; you have to count the 36 ordered pairs.

A probability is always between 0 (impossible) and 1 (certain).

Sample spaces to know

Cards in detail: hearts and diamonds are red (26 cards); spades and clubs are black (26 cards). Each suit has an ace, 2 to 10, and three face cards (jack, queen, king). So there are 4 aces, 4 kings and 12 face cards in all.

Two dice, ways to get each sum:

The pattern rises to 6 at a sum of 7 and falls again, so you can rebuild it in seconds.

Four tools

• Complement: P(not A) = 1 − P(A). For "at least one", find the chance of "none" and subtract from 1. It is almost always shorter.
• Overlap: P(A or B) = P(A) + P(B) − P(A and B). When you count "red or king", the two red kings are in both groups; count them once.
• Choosing together: when r items are drawn together from n, the number of ways is nCr = n! ÷ (r! × (n − r)!). Useful values: 5C2 = 10, 8C2 = 28, 10C2 = 45, 12C3 = 220.
• One after another: for independent events, multiply. With replacement the bag is the same each time. Without replacement, reduce the counts after each draw.

Worked examples

Example 1. A die is thrown. What is the probability of a number greater than 4?

1. Favourable: 5 and 6, so 2 outcomes out of 6.
2. P = 2/6 = 1/3.

Example 2. Two dice are thrown. What is the probability that the sum is 7? That the sum is 8?

1. Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so 6 ways. P = 6/36 = 1/6.
2. Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2), so 5 ways. P = 5/36.

Example 3. Three coins are tossed. What is the probability of at least one head?

1. Total = 8. "No head" = TTT, only 1 outcome.
2. P(at least one head) = 1 − 1/8 = 7/8.

Example 4. A card is drawn from a pack of 52. What is the probability that it is red or a king?

1. Red cards = 26. Kings = 4. Red kings = 2, counted in both.
2. Favourable = 26 + 4 − 2 = 28. P = 28/52 = 7/13.

Example 5. A bag has 4 red and 6 blue balls. Two balls are drawn together. Find the probability that (a) both are red, (b) one is red and one is blue, (c) at least one is red.

1. Total ways = 10C2 = 45.
2. (a) 4C2 = 6, so P = 6/45 = 2/15.
3. (b) 4 × 6 = 24, so P = 24/45 = 8/15.
4. (c) Both blue = 6C2 = 15, so P = 1 − 15/45 = 30/45 = 2/3.
5. Check: (a) + (b) + both blue = 6 + 24 + 15 = 45. Every case is covered once.

Example 6. A bag has 3 white and 2 black balls. Two balls are drawn one after the other. Find the probability that both are white (a) with replacement, (b) without replacement.

1. (a) 3/5 × 3/5 = 9/25.
2. (b) After one white ball is taken, 2 white remain out of 4. So 3/5 × 2/4 = 3/10.
3. Check (b) the "together" way: 3C2 ÷ 5C2 = 3/10. The same.

Common mistakes

• Treating the 11 possible sums of two dice as equally likely. Count the 36 pairs.
• Counting (2, 5) and (5, 2) as one outcome. With two dice they are different outcomes.
• Adding "red" and "king" without removing the red kings.
• Forgetting to reduce the counts in a draw without replacement.
• Working out "at least one" case by case when 1 − P(none) takes one line.

Practice

Set a six-minute timer.

1. A die is thrown. What is the probability of a multiple of 3?
2. Two dice are thrown. What is the probability that the sum is 10?
3. A bag has 3 red and 5 green balls. One ball is drawn. What is the probability that it is green?
4. From the same bag, two balls are drawn together. What is the probability that both are green?
5. Two dice are thrown. What is the probability of a doublet (the same number on both)?
6. A card is drawn from a pack of 52. What is the probability that it is a spade or an ace?
7. Two coins are tossed. What is the probability of exactly one head?
8. A bag has 5 red, 4 blue and 3 green balls. Three balls are drawn together. What is the probability that they are all of different colours?

Answers:

1. 1/3. 3 and 6: 2 out of 6.
2. 1/12. (4,6), (5,5), (6,4): 3 out of 36.
3. 5/8.
4. 5/14. 5C2 ÷ 8C2 = 10/28.
5. 1/6. Six doublets out of 36.
6. 4/13. 13 spades + 4 aces − 1 ace of spades = 16. 16/52 = 4/13.
7. 1/2. HT and TH: 2 out of 4.
8. 3/11. One of each colour = 5 × 4 × 3 = 60 ways. Total = 12C3 = 220. 60/220 = 3/11.

What to do next

• Write out the two-dice sum table from memory until it takes under a minute.
• Learn the card facts in the table: 26 red, 26 black, 12 face cards, 4 of each rank.
• Practise 10 bag-of-balls questions using nCr, then redo two of them the one-by-one way to confirm they match.
• Probability fractions often need quick simplifying; faster calculation helps. For counting habits in longer questions, see word problems.