Probability — Study Notes for IBPS Clerk Prelims
Overview
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain).
The exam focuses on single-event probability involving coins, dice, balls drawn from bags, and cards drawn from a deck. You won't encounter conditional probability or complex multi-event scenarios at the Clerk level. The key is to accurately count favorable outcomes and total outcomes, then apply the basic formula without making careless errors.
This topic rewards students who memorize standard outcomes (36 combinations for two dice, 52 cards in a deck) and practice enough to recognize patterns instantly. Time saved here can be invested in lengthy DI or puzzle questions.
Key Concepts
- Probability Formula: P(Event) = Number of favorable outcomes ÷ Total number of possible outcomes. This single formula drives every Clerk-level problem.
- Sample Space: The set of all possible outcomes. For one coin: {H, T} = 2 outcomes. For one die: {1, 2, 3, 4, 5, 6} = 6 outcomes.
- Favorable Outcomes: Outcomes that satisfy the condition asked in the question. Count these carefully — most errors happen here.
- Complementary Events: P(Event not happening) = 1 − P(Event happening). Useful when counting "at least one" scenarios.
- Independent Events: When two events don't affect each other (like tossing two coins), multiply their individual probabilities: P(A and B) = P(A) × P(B).
- Mutually Exclusive Events: Events that cannot happen together (like getting both head and tail on one coin toss). For these: P(A or B) = P(A) + P(B).
- Equally Likely Outcomes: The probability formula works only when all outcomes have equal chance of occurring — standard assumption in exam problems.
Formulas / Key Facts
Core Formula P(E) = Favorable outcomes / Total outcomes
Standard Sample Spaces — Memorize These
| Experiment | Total Outcomes |
|---|---|
| 1 coin tossed | 2 |
| 2 coins tossed | 4 |
| 3 coins tossed | 8 |
| n coins tossed | 2ⁿ |
| 1 die rolled | 6 |
| 2 dice rolled | 36 |
| 1 card drawn from deck | 52 |
| 1 ball drawn from bag of n balls | n |
Playing Cards Breakdown (52 cards)
- 4 suits: Spades (♠), Hearts (♥), Diamonds (♦), Clubs (♣)
- Each suit: 13 cards (A, 2–10, J, Q, K)
- Red cards: 26 (Hearts + Diamonds)
- Black cards: 26 (Spades + Clubs)
- Face cards: 12 (4 Jacks + 4 Queens + 4 Kings)
- Aces: 4
Dice Outcomes for Two Dice
- Sum of 7: 6 ways → (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum of 2: 1 way → (1,1)
- Sum of 12: 1 way → (6,6)
- Doublets: 6 ways → (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
Worked Examples
Example 1: Single Die A die is rolled once. What is the probability of getting a number greater than 4?
Total outcomes = 6 (numbers 1 to 6) Favorable outcomes = {5, 6} = 2 P = 2/6 = 1/3
Example 2: Two Coins Two coins are tossed simultaneously. Find the probability of getting at least one head.
Total outcomes = 4 → {HH, HT, TH, TT} Favorable (at least one H) = {HH, HT, TH} = 3 P = 3/4
Alternative using complement: P(at least one H) = 1 − P(no heads) = 1 − P(TT) = 1 − 1/4 = 3/4
Example 3: Balls in a Bag A bag contains 5 red balls, 4 blue balls, and 3 green balls. One ball is drawn at random. What is the probability that it is not green?
Total balls = 5 + 4 + 3 = 12 Green balls = 3 Not green = 12 − 3 = 9 P(not green) = 9/12 = 3/4
Example 4: Playing Cards One card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing a face card.
Total cards = 52 Face cards = 12 (J, Q, K in each of 4 suits) P = 12/52 = 3/13
Example 5: Two Dice Two dice are rolled together. What is the probability that the sum is 9?
Total outcomes = 36 Favorable combinations for sum 9: (3,6), (4,5), (5,4), (6,3) = 4 ways P = 4/36 = 1/9
Common Mistakes
- Confusing "at least one" with "exactly one" → "At least one head in 2 tosses" includes HH, HT, TH (3 outcomes), while "exactly one head" includes only HT, TH (2 outcomes). Read the question word precisely.
- Miscounting two-dice combinations → Students count (3,4) and (4,3) as one outcome. They are different! Two dice give 36 ordered pairs, not 21 unordered combinations.
- Forgetting that Ace is not a face card → Face cards are only Jack, Queen, King. Aces are numbered cards. Face cards = 12, not 16.
- Adding probabilities when you should multiply → For "coin shows head AND die shows 6" (both events together), multiply: P = 1/2 × 1/6 = 1/12. Addition is for "either-or" with mutually exclusive events.
- Wrong total count in ball problems → Always add all balls in the bag first. If the problem says "5 red and 7 blue balls," total = 12, not 5 or 7.
- Ignoring "without replacement" vs "with replacement" → At Clerk level, single-draw problems dominate, so this rarely applies. But if two balls are drawn without replacement, the second draw has one fewer ball in total.
Quick Reference
- P(E) = Favorable / Total — the only formula you need for Clerk level.
- 1 die = 6 outcomes; 2 dice = 36 outcomes; 1 coin = 2; 2 coins = 4; n coins = 2ⁿ.
- Deck: 52 cards, 4 suits, 13 each; Face cards = 12; Aces = 4.
- "At least one" problems: use complement method → 1 − P(none).
- Two dice sum of 7 has maximum combinations (6 ways); sums of 2 and 12 have minimum (1 way each).
- Probability always lies between 0 and 1; if your answer exceeds 1, recheck your counting.