Probability Questions for Bank Exams: The Core Formula
Every probability question in bank exams starts from one formula: P(event) = favourable outcomes ÷ total outcomes, and the answer always lies between 0 and 1. For example, the probability of getting a sum of 8 with two dice is 5/36, because 5 of the 36 equally likely pairs, (2,6), (3,5), (4,4), (5,3) and (6,2), give 8.
Key takeaway: Count the sample space first (36 for two dice, 8 for three coins, 52 for cards, nCr for balls), then count favourable cases. For "at least one" use 1 − P(none). For independent events multiply; for mutually exclusive events add.
SBI PO and IBPS PO prelims usually ask one or two probability questions, often inside a "quantity I vs quantity II" or data-sufficiency frame, and mains can ask a full set on balls or cards. Every answer in this note was checked by full enumeration in Python.
Probability Formulas for Bank Exams
| Concept | Formula |
|---|---|
| Basic probability | P(E) = n(E) ÷ n(S) |
| Complement | P(not E) = 1 − P(E) |
| At least one | 1 − P(none) |
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) |
| Mutually exclusive | P(A or B) = P(A) + P(B) |
| Independent events | P(A and B) = P(A) × P(B) |
| Choosing r of n | nCr = n! ÷ [r! × (n − r)!] |
| Odds in favour a : b | P(E) = a ÷ (a + b) |
Combinations do most of the counting, so revise nCr values like 5C2 = 10, 6C2 = 15, 12C2 = 66, 12C3 = 220 and 15C3 = 455 until they are instant. If calculations slow you down, our simplification tricks for bank exams help.
Sample Space Table (Learn Before the Exam)
| Experiment | Total outcomes |
|---|---|
| 1 coin | 2 |
| 2 coins | 4 |
| 3 coins | 8 |
| n coins | 2ⁿ |
| 1 die | 6 |
| 2 dice | 36 |
| 3 dice | 216 |
| 1 card from a deck | 52 |
| r balls from n | nCr |
Deck of cards facts: 52 cards, 4 suits of 13 each (spades and clubs black, hearts and diamonds red), 4 aces, 4 kings, 4 queens, 4 jacks, so 12 face cards (16 if aces are counted as honours, so read the question).
Two Dice Sum Table (Most Asked Shortcut)
| Sum | Ways | Probability |
|---|---|---|
| 2 or 12 | 1 | 1/36 |
| 3 or 11 | 2 | 1/18 |
| 4 or 10 | 3 | 1/12 |
| 5 or 9 | 4 | 1/9 |
| 6 or 8 | 5 | 5/36 |
| 7 | 6 | 1/6 |
The pattern rises by one way per step up to 7 and then falls. Doublets (same number on both dice) are 6 of 36, that is 1/6.
Probability Questions on Dice, Coins and Cards (Solved)
Q1. Two dice are thrown. Find the probability that the sum is 8. 5 favourable pairs out of 36: 5/36.
Q2. Three coins are tossed. Find the probability of at least two heads. HHT, HTH, THH, HHH = 4 of 8: 1/2.
Q3. One card is drawn from a deck. Find the probability that it is a king or a heart. 4 kings + 13 hearts − 1 king of hearts = 16. P = 16/52 = 4/13.
Q4. Two dice are thrown. Find the probability of a doublet or a sum of 10. Doublets 6, sum 10 gives 3 pairs, (5,5) is counted twice, so 6 + 3 − 1 = 8. P = 8/36 = 2/9.
Q5. Two cards are drawn without replacement. Find the probability that both are aces. 4/52 × 3/51 = 1/221.
Probability Questions on Balls (SBI PO Pattern)
Use this bag for Q6 to Q8: 5 red, 4 blue and 3 green balls (12 in all).
Q6. Two balls are drawn at random. Find the probability that both are red. 5C2 ÷ 12C2 = 10/66 = 5/33.
Q7. Two balls are drawn. Find the probability that they are of different colours. Same colour = (5C2 + 4C2 + 3C2) ÷ 12C2 = (10 + 6 + 3)/66 = 19/66. Different = 1 − 19/66 = 47/66.
Q8. Three balls are drawn. Find the probability that at least one is green. None green = 9C3 ÷ 12C3 = 84/220. At least one = 1 − 84/220 = 136/220 = 34/55.
Q9. A bag has 5 red balls and some blue balls. The probability of drawing a blue ball is twice that of a red ball. How many blue balls are there? x/(5 + x) = 2 × 5/(5 + x), so x = 10.
Independent Events and At Least One (Solved)
Q10. A can solve a problem with probability 1/3 and B with probability 1/4. If both try independently, find the probability that the problem is solved. P(neither) = 2/3 × 3/4 = 1/2. P(solved) = 1 − 1/2 = 1/2.
Q11. A number is chosen from 1 to 50. Find the probability that it is divisible by 4 or 5. Multiples of 4: 12, of 5: 10, of 20: 2. Count = 12 + 10 − 2 = 20. P = 20/50 = 2/5.
Q12. A committee of 3 is chosen from 4 men and 3 women. Find the probability that exactly 2 are women. (3C2 × 4C1) ÷ 7C3 = 12/35 = 12/35.
The counting in Q12 is the same nCr method you use in seating and selection puzzles; see our circular seating arrangement tricks for the reasoning side.
Probability Short Tricks for SBI PO and IBPS PO
- "At least one" → 1 − none. Faster than adding every case (Q8, Q10).
- "Or" with overlap → subtract the common part. King or heart, doublet or sum 10 (Q3, Q4).
- Without replacement → reduce both numerator and denominator at the second draw (Q5).
- Same colour vs different colour: find "same", then subtract from 1 (Q7).
- Check the range: a probability above 1 or below 0 means a counting error.
- Data sufficiency style: if a statement gives the total and one probability, you can usually find the missing count (Q9).
Common Mistakes in Probability Questions
| Mistake | Correct approach |
|---|---|
| Counting (2,6) and (6,2) as one outcome | Dice are distinct, so both count |
| Using 13 face cards | There are 12 face cards (J, Q, K of four suits) |
| Adding probabilities of overlapping events | Subtract P(A and B) |
| Multiplying events that are not independent | Without replacement, change the second fraction |
| Counting a leap year as 1/7 for 53 Sundays | A leap year has 2 extra days, so 2/7 |
Probability in Quantity I vs Quantity II Questions (SBI PO Format)
SBI PO and IBPS PO prelims often hide probability inside a quantity comparison. Work out both quantities as fractions and compare.
Example A. Quantity I: probability of getting at least one six when a die is thrown twice. Quantity II: probability of getting a sum of 7 with two dice. Quantity I = 1 − (5/6)² = 11/36. Quantity II = 6/36. So Quantity I > Quantity II.
Example B. A bag has 4 red and 6 blue balls and two are drawn. Quantity I: probability that both are blue. Quantity II: probability of one red and one blue. Quantity I = 6C2 ÷ 10C2 = 15/45 = 1/3. Quantity II = (4 × 6) ÷ 45 = 24/45 = 8/15. So Quantity II > Quantity I.
Tip: keep both answers over the same denominator (36 for dice, 10C2 = 45 here) and compare numerators; there is no need to simplify.
Probability Quick Revision Table
| Question type | First step | Shortcut |
|---|---|---|
| Two dice, given sum | Use the sum table | 7 is most likely (6/36) |
| Coins | Total 2ⁿ | Exactly r heads = nCr ÷ 2ⁿ |
| Cards | Total 52 | Watch overlaps (king of hearts) |
| Balls, same colour | Add nCr for each colour | Different colour = 1 − same |
| At least one | P(none) | 1 − P(none) |
| Two people, independent | Multiply | Solved = 1 − both fail |
Probability Practice Questions for Bank Exams (with Answers)
- Two dice are thrown. Probability that the sum is a prime number? Answer: 5/12 (sums 2, 3, 5, 7, 11 → 15 ways).
- Four coins are tossed. Probability of exactly 2 heads? Answer: 3/8 (4C2 = 6 of 16).
- One card is drawn. Probability that it is a face card? Answer: 3/13 (12/52).
- A bag has 6 white and 4 black balls. Two are drawn. Probability of one of each colour? Answer: 8/15 (24/45).
- A number from 1 to 30 is chosen. Probability that it is prime? Answer: 1/3 (10 primes).
- Probability that a leap year has 53 Sundays? Answer: 2/7.
- Two dice are thrown. Probability that the product is even? Answer: 3/4 (1 − 9/36).
- A hits a target with probability 0.6 and B with 0.5. Probability that exactly one hits? Answer: 1/2 (0.6 × 0.5 + 0.4 × 0.5).
- A bag has 4 red, 5 blue and 6 green balls. Three are drawn. Probability that all three are the same colour? Answer: 34/455 (4 + 10 + 20 = 34 of 455).
- Two dice are thrown. Probability that the sum is at least 10? Answer: 1/6 (6 of 36).
Put this into timed practice with the SBI PO Quantitative Aptitude Sectional Mock Test 02 and the IBPS PO Quantitative Aptitude Sectional Mock Test 03.
Probability for Bank Exams FAQs
How many probability questions come in SBI PO?
Usually one or two in prelims, often as part of a quantity comparison or data sufficiency question, and sometimes a short set in mains.
What is the formula of probability?
P(E) = number of favourable outcomes ÷ total number of outcomes. It always lies between 0 and 1.
How do you solve "at least one" probability questions?
Find the probability that none of the events happens and subtract it from 1.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other, so you multiply their probabilities. Mutually exclusive events cannot happen together, so you add their probabilities.
Do I need permutation and combination for probability?
Yes. Ball, card and committee questions all need nCr to count the sample space and favourable cases.
Practice Next
- Mock tests: SBI PO Quantitative Aptitude Sectional Mock Test 04 and IBPS Clerk Numerical Ability Sectional Mock Test 04.
- Related notes: mixture and alligation questions for bank exams, quadratic equation tricks for bank exams, wrong number series for bank exams, calendar odd days trick and permutation and combination for bank exams.
- Exam details: SBI Clerk 2026 notification.
- Browse all study notes.