Probability with AND and OR

Review: Basic Probability

Probability: P(event) = (favorable outcomes)/(total outcomes)

Range: 0 ≤ P ≤ 1

Certain event: P = 1

Impossible event: P = 0

Compound Events

Compound event: Combination of two or more simple events

Two types:

• AND: Both events happen
• OR: At least one event happens

The AND Rule (Multiplication)

P(A AND B): Probability both A and B occur

For independent events:

P(A AND B) = P(A) × P(B)

Independent: One event doesn't affect the other

Example 1: Two Coin Flips

Flip coin twice. Probability of heads both times?

P(heads first) = 1/2 P(heads second) = 1/2

P(heads AND heads):

= 1/2 × 1/2
= 1/4

Answer: 1/4 or 0.25

Example 2: Rolling Dice

Roll two dice. Probability both show 6?

P(6 on first die) = 1/6 P(6 on second die) = 1/6

P(both 6s):

= 1/6 × 1/6
= 1/36

Answer: 1/36

Example 3: Drawing with Replacement

Bag: 3 red, 2 blue marbles. Draw one, replace, draw again. P(red then blue)?

P(red first) = 3/5 P(blue second) = 2/5 (replaced, same total)

P(red AND blue):

= 3/5 × 2/5
= 6/25

Answer: 6/25

Dependent Events

Dependent: First event affects probability of second

For dependent events:

P(A AND B) = P(A) × P(B|A)

Where P(B|A) = probability of B given A occurred

Example 1: Drawing Without Replacement

Bag: 3 red, 2 blue marbles. Draw two without replacing. P(both red)?

First draw:

P(red) = 3/5

Second draw (given first was red):

P(red | first red) = 2/4 = 1/2
(only 2 red left out of 4 total)

P(red AND red):

= 3/5 × 1/2 = 3/10

Answer: 3/10

Example 2: Cards Without Replacement

Draw 2 cards from standard deck. P(both aces)?

First card:

P(ace) = 4/52 = 1/13

Second card (given first was ace):

P(ace | first ace) = 3/51 = 1/17

P(ace AND ace):

= 1/13 × 1/17 = 1/221

Answer: 1/221

The OR Rule (Addition)

P(A OR B): Probability of A, or B, or both

For mutually exclusive events (cannot both happen):

P(A OR B) = P(A) + P(B)

Example 1: Mutually Exclusive

Roll die. P(3 or 5)?

Cannot roll both at once (mutually exclusive)

P(3) = 1/6 P(5) = 1/6

P(3 OR 5):

= 1/6 + 1/6 = 2/6 = 1/3

Answer: 1/3

Example 2: Card Draw

Draw one card. P(ace or king)?

Cannot be both (mutually exclusive)

P(ace) = 4/52 P(king) = 4/52

P(ace OR king):

= 4/52 + 4/52 = 8/52 = 2/13

Answer: 2/13

Non-Mutually Exclusive Events

Can happen together: Need to subtract overlap

General OR rule:

P(A OR B) = P(A) + P(B) - P(A AND B)

Subtract overlap to avoid counting twice

Example 1: Overlapping Events

Draw card. P(heart OR face card)?

P(heart) = 13/52 P(face card) = 12/52 P(heart AND face) = 3/52 (Jack, Queen, King of hearts)

P(heart OR face):

= 13/52 + 12/52 - 3/52
= 22/52
= 11/26

Answer: 11/26

Example 2: Rolling Dice

Roll die. P(even OR greater than 3)?

Even: {2, 4, 6} → P(even) = 3/6 Greater than 3: {4, 5, 6} → P(>3) = 3/6 Both: {4, 6} → P(even AND >3) = 2/6

P(even OR >3):

= 3/6 + 3/6 - 2/6
= 4/6
= 2/3

Answer: 2/3

At Least One

"At least one": Use complement

P(at least one) = 1 - P(none)

Example 1: Coin Flips

Flip coin 3 times. P(at least one heads)?

P(no heads) = P(all tails):

= 1/2 × 1/2 × 1/2 = 1/8

P(at least one heads):

= 1 - 1/8 = 7/8

Answer: 7/8

Example 2: Defective Items

3 items, each 10% defective. P(at least one works)?

P(one defective) = 0.1 P(one works) = 0.9

P(all defective):

= 0.1 × 0.1 × 0.1 = 0.001

P(at least one works):

= 1 - 0.001 = 0.999

Answer: 0.999 or 99.9%

Multiple Events

Combine AND and OR rules

Example: Three Coin Flips

P(exactly 2 heads in 3 flips)?

Outcomes with exactly 2 heads:

• HHT
• HTH
• THH

Each has probability:

1/2 × 1/2 × 1/2 = 1/8

Three ways, so:

P(exactly 2 heads) = 3 × 1/8 = 3/8

Answer: 3/8

Tree Diagrams

Visual tool for compound events

Branch for each outcome, multiply along paths

Example: Two Flips

        H (1/2) — HH (1/4)
H (1/2)
        T (1/2) — HT (1/4)

        H (1/2) — TH (1/4)
T (1/2)
        T (1/2) — TT (1/4)

Summary of Rules

Independent AND: P(A AND B) = P(A) × P(B)

Dependent AND: P(A AND B) = P(A) × P(B|A)

Mutually Exclusive OR: P(A OR B) = P(A) + P(B)

General OR: P(A OR B) = P(A) + P(B) - P(A AND B)

Complement: P(not A) = 1 - P(A)

At least one: P(at least one) = 1 - P(none)

Real-World Applications

Quality control: P(multiple defects)

Medical testing: P(disease AND positive test)

Weather: P(rain Monday OR Tuesday)

Games: P(winning multiple rounds)

Insurance: P(multiple claims)

Example: Quality Control

Machine produces 5% defective items. Inspect 2 random items. P(both defective)?

Assuming independent:

P(both defective) = 0.05 × 0.05 = 0.0025

Answer: 0.0025 or 0.25%

Practice