Understanding Probability Addition Theorem: Breaking it down into two rules

                                      

To understand this theorem, split it into two rules:

Rule 1: Mutually exclusive events (cannot occur together).

Rule 2: Non-mutually exclusive events (can overlap).

1. Mutually Exclusive Events

Definition: Two events are mutually exclusive if one prevents the other.

Example 1 – Dice:

When rolling a die, only one face appears. If "5" lands, "3" cannot occur simultaneously.

Rule 1:

              P (A ⋃ B) = P(A) +P (B)

Example 2 – Colored Balls:
A jar has 11 balls (3 blue, 4 white, 2 red, 1 yellow, 1 green).

What’s the probability of drawing green or yellow?

·       P(green) = 1/11

·       P(yellow)=1/11

·       P (green ⋃ yellow) =11/1 + 1/11=2/11

2. Non-Mutually Exclusive Events

Definition: They share at least one common outcome.

Example 3 – Dice:

·        Event A: Odd number (1, 3, 5).

·        Event B: Number >4 (5, 6).

·        Intersection: The number 5 is in both.

Rule 2:

                                   P (A ⋃ B) = P(A) + P(B) −P (A ∩ B)

Example 4 – Deck of Cards:
What’s the probability of drawing a king or a heart?

·        P (king)=4/52

·        ​P (heart)=13/52

·        P (king of hearts) =1/52​.

·        P (king or heart) = 4/52+13/52−1/52=16/52​.

Exercises & Answers

1.     Given A = {1, 2, …,10}:
a) Odd <4 or odd >8 → 3/10
b) Odd or multiple of 3 → 3/5​.

2.     Die: odd or multiple of 3 → 2/3​.

3.     Coin (heads) or die (2) → 7/12​.