🎲 Probability - Complete Theory

Master the mathematics of chance and uncertainty!

🎯 What is Probability?

Probability is the measure of likelihood that an event will occur.

Probability (P) = Number of Favorable Outcomes / Total Number of Possible Outcomes

P(E) = n(E) / n(S)

Where:
E = Event
S = Sample Space (all possible outcomes)

Range: 0 ≤ P(E) ≤ 1

• P = 0 → Impossible event
• P = 1 → Certain event
• P = 0.5 → Equally likely

📐 Basic Formulas

Formula 1: Complementary Events

P(E) + P(not E) = 1
P(not E) = 1 - P(E)

Example: P(rain) = 0.3
P(no rain) = 1 - 0.3 = 0.7

Formula 2: Addition Rule (OR)

For mutually exclusive events (can't happen together):
P(A or B) = P(A) + P(B)

For non-mutually exclusive events:
P(A or B) = P(A) + P(B) - P(A and B)

Formula 3: Multiplication Rule (AND)

For independent events:
P(A and B) = P(A) × P(B)

For dependent events:
P(A and B) = P(A) × P(B|A)
where P(B|A) = probability of B given A occurred

🎴 Standard Probability Scenarios

1. Dice (Single Die)

Sample Space = {1, 2, 3, 4, 5, 6}
Total outcomes = 6

P(getting 4) = 1/6
P(even number) = 3/6 = 1/2  {2, 4, 6}
P(number > 4) = 2/6 = 1/3  {5, 6}
P(prime) = 3/6 = 1/2  {2, 3, 5}

2. Two Dice

Total outcomes = 6 × 6 = 36

P(sum = 7) = 6/36 = 1/6
Combinations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

P(sum = 12) = 1/36
Combination: (6,6)

P(doublet) = 6/36 = 1/6
Combinations: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

3. Playing Cards

Standard Deck:

Total cards = 52
Suits: Hearts ♥, Diamonds ♦ (Red) - 26 cards
       Clubs ♣, Spades ♠ (Black) - 26 cards

Each suit has 13 cards:
A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Face cards (Court cards) = 12 (J, Q, K in each suit)
Number cards = 40
Aces = 4

Common Probabilities:

P(King) = 4/52 = 1/13
P(Red card) = 26/52 = 1/2
P(Spade) = 13/52 = 1/4
P(Face card) = 12/52 = 3/13
P(Ace of Hearts) = 1/52

4. Balls/Marbles in Bag

Example: Bag with 5 red, 3 blue, 2 green balls

Total = 10

P(red) = 5/10 = 1/2
P(blue) = 3/10
P(not green) = 8/10 = 4/5

💡 Solved Examples

Example 1: Basic Probability

Q: A bag has 3 red and 5 black balls. Find probability of drawing a red ball.

Solution:

Total balls = 3 + 5 = 8
Red balls = 3

P(red) = 3/8

Answer: 3/8

Example 2: Complementary Event

Q: Probability of passing exam is 0.75. Find probability of failing.

Solution:

P(fail) = 1 - P(pass)
        = 1 - 0.75
        = 0.25

Answer: 0.25 or 25%

Example 3: Two Dice Sum

Q: Two dice are thrown. Find probability that sum is at least 10.

Solution:

Total outcomes = 36

Sum ≥ 10 means sum = 10, 11, or 12

Sum = 10: (4,6), (5,5), (6,4) → 3 ways
Sum = 11: (5,6), (6,5) → 2 ways
Sum = 12: (6,6) → 1 way

Favorable = 3 + 2 + 1 = 6

P(sum ≥ 10) = 6/36 = 1/6

Answer: 1/6

Example 4: Playing Cards

Q: One card drawn from deck. Find P(King or Queen).

Solution:

Kings = 4
Queens = 4
Total = 4 + 4 = 8 (mutually exclusive)

P(King or Queen) = 8/52 = 2/13

Answer: 2/13

Example 5: Independent Events

Q: Two cards drawn WITH REPLACEMENT. Find P(both are Aces).

Solution:

P(1st Ace) = 4/52 = 1/13
P(2nd Ace) = 4/52 = 1/13 (replacement means deck restored)

P(both Aces) = 1/13 × 1/13 = 1/169

Answer: 1/169

Example 6: Dependent Events

Q: Two cards drawn WITHOUT REPLACEMENT. Find P(both are Kings).

Solution:

P(1st King) = 4/52 = 1/13

After removing 1 King:
Cards left = 51
Kings left = 3

P(2nd King | 1st King) = 3/51 = 1/17

P(both Kings) = 1/13 × 1/17 = 1/221

Answer: 1/221

Example 7: At Least One

Q: Coin tossed 3 times. Find P(at least one head).

Solution:

Method 1: Use complement
P(no heads) = P(all tails) = (1/2)³ = 1/8
P(at least one head) = 1 - 1/8 = 7/8

Method 2: Direct (longer!)
P(1H or 2H or 3H) = ... = 7/8

Answer: 7/8

Example 8: Conditional Probability

Q: Bag has 4 red, 6 blue balls. Two balls drawn without replacement. Find P(2nd is red | 1st is blue).

Solution:

Given 1st is blue:
Balls left = 9
Red balls left = 4 (unchanged)

P(2nd red | 1st blue) = 4/9

Answer: 4/9

🎯 Important Probability Patterns

Pattern 1: Coin Tosses

For n coins (or n tosses):
Total outcomes = 2ⁿ

1 coin: 2 outcomes (H, T)
2 coins: 4 outcomes (HH, HT, TH, TT)
3 coins: 8 outcomes

Pattern 2: At Least/At Most

"At least one" = 1 - P(none)
"At most one" = P(zero) + P(one)

This is usually easier than direct calculation!

Pattern 3: Dice Sum Probabilities

Most likely sum with 2 dice = 7 (6 ways)
Least likely sums = 2 and 12 (1 way each)

Sum frequency forms a triangle:
Sum 7: ●●●●●● (6 ways)
Sum 6,8: ●●●●● (5 ways each)
Sum 5,9: ●●●● (4 ways each)
...

⚡ Quick Shortcuts

Shortcut 1: Deck Probabilities

P(specific card) = 1/52
P(specific rank) = 4/52 = 1/13
P(specific suit) = 13/52 = 1/4
P(red/black) = 26/52 = 1/2
P(face card) = 12/52 = 3/13

Shortcut 2: Dice Complement

P(at least one 6 in n dice) = 1 - (5/6)ⁿ

For 2 dice: 1 - (5/6)² = 1 - 25/36 = 11/36

Shortcut 3: Same Color Balls

If r red and b black balls, drawing 2:
P(both same color) = [r(r-1) + b(b-1)] / [(r+b)(r+b-1)]

Shortcut 4: Exactly k Successes

Use binomial: C(n,k) × p^k × (1-p)^(n-k)
where n = trials, k = successes, p = probability

📊 Special Cases

Birthday Paradox

Probability that 2 people in group of n share birthday:
P ≈ 1 - (365/365 × 364/365 × 363/365 × ... for n terms)

For 23 people: P ≈ 50.7% (surprising!)

Gambler’s Fallacy

Wrong thinking: "I got 5 tails, next must be heads!" ✗
Right: Each toss is independent, P(H) = 0.5 always ✓

⚠️ Common Mistakes

❌ Mistake 1: Replacement Confusion

WITH replacement: Probability stays same
WITHOUT replacement: Probability changes

Always check question carefully!

❌ Mistake 2: Or vs And

Wrong: P(A or B) = P(A) × P(B) ✗
Right:
  "OR" → Add (check for overlap!)
  "AND" → Multiply ✓

❌ Mistake 3: At Least One

Wrong: Direct calculation (tedious) ✗
Right: Use complement: 1 - P(none) ✓

❌ Mistake 4: Independent Assumption

Wrong: Always using P(A and B) = P(A) × P(B) ✗
Right: Only for independent events ✓
Cards without replacement → NOT independent!

❌ Mistake 5: Favorable Counting

Wrong: P(King or Queen) = 4/52 + 4/52 = 8/52 = 2/13 ✓ (correct here!)
But if overlapping events, must subtract P(both)!

📝 Practice Problems

Level 1:

1. A die is thrown. Find P(getting prime number).
2. A coin is tossed twice. Find P(getting at least one head).
3. One card from deck. Find P(Ace).

Level 2:

4. Two dice thrown. Find P(sum = 8).
5. Bag has 5 red, 3 blue balls. Two drawn without replacement. Find P(both red).
6. Three coins tossed. Find P(exactly 2 heads).

Level 3:

7. A die thrown 3 times. Find P(getting at least one 6).
8. From deck, 3 cards drawn without replacement. Find P(all are Kings).
9. Bag has 4 red, 6 black, 5 white balls. Find P(drawing red and then black without replacement).

🔗 Related Topics

Prerequisites:

• Permutation & Combination - For counting favorable outcomes
• Ratio & Proportion - Understanding ratios in probability

Related:

• Percentage - Converting probability to percentage

Practice:

• Probability Questions
• Formula Sheet

Master Probability - Think in fractions and use complements! 🎲