Probability

Probability — SBI Clerk Prelims Study Notes

Overview

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). Questions are calculation-light but concept-dependent; a clear understanding of basic definitions gets you quick marks.

This topic connects naturally with Permutation and Combination, but at Clerk level, you rarely need advanced counting. Most problems use direct formulas with small, countable outcomes. Master the core formula, learn to identify favourable vs total outcomes, and you'll solve these in under 30 seconds each.

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Key Concepts

• Experiment and Outcome: An experiment is any action with uncertain results (tossing a coin). Each possible result is an outcome (Head or Tail).
• Sample Space (S): The set of all possible outcomes. For one die, S = {1, 2, 3, 4, 5, 6}, so n(S) = 6.
• Event (E): A subset of the sample space that we care about. "Getting an even number" means E = {2, 4, 6}.
• Favourable Outcomes: Outcomes that satisfy the event condition. Count them carefully — this is where most errors happen.
• Probability Range: Always between 0 and 1 inclusive. If your answer exceeds 1 or is negative, recheck immediately.
• Complementary Events: P(Event not happening) = 1 − P(Event happening). Useful when "at least one" type wording appears.
• Equally Likely Outcomes: Probability formulas assume each outcome has the same chance. A fair coin, fair die, or well-mixed bag satisfies this.
• Mutually Exclusive Events: Events that cannot occur together. P(A or B) = P(A) + P(B) when A and B are mutually exclusive.

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Formulas / Key Facts

Core Formula

P(E) = Number of favourable outcomes / Total number of outcomes
     = n(E) / n(S)

Standard Sample Spaces

Coin Facts

• P(Head) = P(Tail) = 1/2
• For n coins: P(exactly k heads) requires combination counting (rare at Clerk level)

Dice Facts

• P(any specific number) = 1/6
• P(even number) = P(odd number) = 3/6 = 1/2
• P(prime number on die) = 3/6 = 1/2 (primes: 2, 3, 5)
• P(number > 4) = 2/6 = 1/3 (numbers: 5, 6)

Ball/Bag Facts

• Add all balls to get total outcomes
• Favourable = count of balls matching the condition
• For "at least one" problems: P(at least one) = 1 − P(none)

Complement Rule

P(not E) = 1 − P(E)

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Worked Examples

Example 1: Single Die

Problem: A fair die is rolled once. What is the probability of getting a number divisible by 3?

Solution:

• Total outcomes = 6 (numbers 1 to 6)
• Favourable outcomes = numbers divisible by 3 = {3, 6} = 2 outcomes
• P(divisible by 3) = 2/6 = 1/3

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Example 2: Bag of Balls

Problem: A bag contains 4 red balls, 5 blue balls, and 3 green balls. One ball is drawn at random. What is the probability that it is not blue?

Solution:

• Total balls = 4 + 5 + 3 = 12
• Blue balls = 5
• Not blue = Red + Green = 4 + 3 = 7
• P(not blue) = 7/12

Alternative using complement: P(not blue) = 1 − P(blue) = 1 − 5/12 = 7/12

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Example 3: Two Coins

Problem: Two fair coins are tossed simultaneously. What is the probability of getting exactly one head?

Solution:

• Sample space for 2 coins = {HH, HT, TH, TT} = 4 outcomes
• Exactly one head = {HT, TH} = 2 outcomes
• P(exactly one head) = 2/4 = 1/2

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Example 4: Cards (Basic)

Problem: One card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is a face card?

Solution:

• Total cards = 52
• Face cards = Jack, Queen, King in each of 4 suits = 3 × 4 = 12
• P(face card) = 12/52 = 3/13

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Common Mistakes

Mistake 1: Miscounting total outcomes

• Wrong thinking: Forgetting to add all categories in a bag problem
• Correct fix: Always write Total = (count each type) and add before calculating

Mistake 2: Confusing "or" with "and"

• Wrong thinking: Adding probabilities when events happen together
• Correct fix: "Or" means add (for mutually exclusive); "and" means multiply (for independent events) — but SBI Clerk rarely tests "and"

Mistake 3: Counting 1 as a prime number

• Wrong thinking: Including 1 in primes when asked "prime number on a die"
• Correct fix: 1 is NOT prime. Primes on a die are 2, 3, 5 only (3 numbers)

Mistake 4: Forgetting to simplify the fraction

• Wrong thinking: Writing 4/12 as final answer
• Correct fix: Always reduce to lowest terms (4/12 = 1/3). Options are usually in simplest form

Mistake 5: Not reading "not" or "at least"

• Wrong thinking: Calculating P(blue) when asked for P(not blue)
• Correct fix: Underline key words in the question before solving

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Quick Reference

• P(E) = Favourable / Total — the only formula you truly need
• Die: 6 outcomes; primes = {2, 3, 5}; evens = {2, 4, 6}
• 2 Coins: 4 outcomes — HH, HT, TH, TT
• Deck of 52: 4 suits × 13 cards; 12 face cards; 4 aces
• Complement shortcut: P(at least one) = 1 − P(none)
• Answer check: If probability > 1 or < 0, you made a counting error