10  Probability

10.1 Introduction

Probability measures the likelihood of an event occurring.
This chapter covers classical and modern definitions, rules of probability, conditional probability, Bayes’ theorem, and practical applications in exams.

We also connect probability with permutations–combinations, since most problems involve counting favorable and total outcomes.

10.2 1) Basic Definitions

  • Experiment: Any action with uncertain outcome (e.g., tossing coin, rolling dice).
  • Sample Space (S): Set of all possible outcomes.
  • Event (E): Subset of sample space.
  • Favorable outcomes: Outcomes consistent with event.

10.2.1 Classical probability

If all outcomes are equally likely:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Example: Tossing a coin, P(Head) = 1/2.

10.3 2) Properties of Probability

  • \(0 \leq P(E) \leq 1\)
  • \(P(S) = 1\), \(P(\emptyset)=0\)
  • Complement rule: \(P(E') = 1 - P(E)\)
  • Addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

10.4 3) Independent and Mutually Exclusive Events

  • Mutually exclusive: \(P(A \cap B)=0\).
  • Independent: \(P(A \cap B) = P(A)\cdot P(B)\).

Be careful: mutually exclusive ≠ independent.

10.5 4) Conditional Probability

Probability of A given B:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B)>0 \]

Multiplication rule:
\(P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)\)

10.6 5) Bayes’ Theorem

For mutually exclusive and exhaustive events \(B_1,B_2,\dots,B_n\):

\[ P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\sum_{j=1}^n P(B_j)P(A|B_j)} \]

Useful in diagnostic tests, reliability, classification.

10.7 6) Classical Models

10.7.1 6.1 Coins

n tosses → \(2^n\) outcomes.
Example: Toss 3 coins, probability of exactly 2 heads = \(3/8\).

10.7.2 6.2 Dice

One die: 6 outcomes. Two dice: 36 outcomes.
Example: Sum = 7 → 6 favorable → 6/36 = 1/6.

10.7.3 6.3 Cards

52-card deck.
- Hearts/diamonds/clubs/spades = 13 each.
- Face cards = 12.
Example: Probability of red card = 26/52=1/2.

10.8 7) Probability using Combinations

Favorable / Total often reduced to nCr ratios.

Example: Probability of drawing 2 aces from 52 cards:
Favorable = 4C2=6, Total = 52C2=1326 ⇒ 6/1326 = 1/221.

10.9 8) Conditional Applications

10.9.1 8.1 Drawing without replacement

Probability changes after each draw.
Example: 2 balls from 5 red + 3 blue.
P(Red, then Blue) = (5/8)(3/7)=15/56.

10.9.2 8.2 Drawing with replacement

Probabilities remain constant.
Same example → (5/8)(3/8)=15/64.

10.10 9) Independent Trials (Bernoulli Model)

Probability of exactly r successes in n independent trials (each success probability p):

\[ P(X=r) = {n \choose r} p^r (1-p)^{n-r} \]

This is the Binomial distribution.

10.11 10) Solved Examples

Ex 1: Probability of getting at least one head in 2 tosses.
= 1 − P(no head) = 1 − (1/2)^2 = 3/4.

Ex 2: A bag has 5 red, 4 black balls. Draw 2 without replacement.
P(both red) = (5/9)(4/8)=20/72=5/18.

Ex 3: Two dice thrown. Probability that sum is even?
Half outcomes even → 18/36 = 1/2.

Ex 4: From 52 cards, probability of king or queen.
= (4+4)/52=8/52=2/13.

Ex 5: Conditional probability:
2 balls drawn without replacement from 3 red + 2 blue. Find P(both red | first red).
P= (2/4)=1/2.

Ex 6: Bayes example:
A factory has 3 machines A,B,C producing 30%,50%,20% of total. Defect rate: 2%,3%,4%. If a product is defective, probability it came from C?
= (0.2×0.04)/(0.3×0.02+0.5×0.03+0.2×0.04)=0.008/0.027=0.296≈29.6%.

10.12 11) Common Traps

  • Forgetting replacement vs no replacement.
  • Confusing mutually exclusive vs independent.
  • Not reducing fractions.
  • Ignoring complement method (often faster).
  • Forgetting denominators shrink in successive draws without replacement.

10.13 Practice Set – Level 1

  1. Toss 2 dice. Probability that sum=11.
  2. Draw 1 card. Probability it is not a heart.
  3. Toss 3 coins. Probability at least 2 heads.
  4. Draw 2 cards from pack. Probability both black.
  5. Roll a die. Probability prime number.

10.14 Practice Set – Level 2

  1. Bag with 6 white, 4 black. Draw 2 without replacement. Probability of same color.
  2. A coin tossed 5 times. Probability exactly 3 heads.
  3. Draw 2 cards. Probability of at least 1 ace.
  4. 3 dice rolled. Probability that sum=9.
  5. A bag has 3 red, 2 blue, 1 green. Draw 2 balls. Probability different colors.

10.15 Practice Set – Level 3 (Challengers)

  1. 10 coins tossed. Probability exactly 5 heads.
  2. A card drawn, replaced, then another. Probability both kings.
  3. Bag with 4 red, 5 black. A ball drawn, found red, not replaced. Then second drawn. Probability black?
  4. Probability that a leap year has 53 Sundays.
  5. In exam, student guesses 6 true/false questions. Probability of exactly 4 correct.

10.16 Answer Key (outline)

Level 1: (1) 2/36=1/18; (2) 39/52=3/4; (3) 4/8=1/2; (4) 26C2/52C2=325/1326=25/102; (5) 3/6=1/2.
Level 2: (6) (6C2+4C2)/10C2=15+6/45=21/45=7/15; (7) 10C3/32=10/32=0.3125; (8) 1−48C2/52C2=1−1128/1326=198/1326=33/221; (9) 25/216; (10) (3×2+3×1+2×1)/15=8/15.
Level 3: (11) 10C5/2^10=252/1024=0.246; (12) (4/52)(4/52)=1/169; (13) 5/8; (14) 2/7; (15) 6C4/2^6=15/64.

10.17 Summary

  • Probability = favorable / total outcomes.
  • Use addition/multiplication rules carefully.
  • Distinguish independent vs mutually exclusive.
  • Master conditional probability and Bayes.
  • Apply binomial for multiple trials.
    Practice ensures quick recognition of models (coins, dice, cards, bags, etc.) in exams.