Addition Rule in Probability: Mutually Exclusive and Non-Mutually Exclusive Events

Probability: Addition Rule

Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot.

• Definition: Two events are mutually exclusive if they have no outcomes in common.

• Example: Getting heads when flipping a coin and getting tails are mutually exclusive events.

• Non-Example: Getting a 4 when rolling a die and getting a number higher than 3 are not mutually exclusive, since 4 is higher than 3.

Key Point: For mutually exclusive events A and B:

• The probability of A or B occurring is the sum of their individual probabilities.

Formula:

$P(A \cup B) = P(A) + P(B)$

Example: In a six-sided die, what is the probability of getting a 3 or 5?

• There are 6 outcomes, and only one way to get a 3 and one way to get a 5.

• $P(3 \text{ or } 5) = P(3) + P(5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = 0.33$

Practice: If a single card is randomly selected from a deck of cards, what is the probability of selecting an ace or a king?

• There are 4 aces and 4 kings in a deck of 52 cards.

• $P(\text{ace or king}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = 0.15$

Non-Mutually Exclusive Events

Non-mutually exclusive events are events that can occur at the same time. There is overlap between the events.

• Definition: Two events are not mutually exclusive if they share at least one outcome.

• Example: Rolling a die: the event "rolling a number greater than 1" and "rolling an even number" overlap at 2, 4, and 6.

Key Point: For non-mutually exclusive events A and B:

• The probability of A or B occurring is the sum of their individual probabilities minus the probability that both occur (the overlap).

Formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Example: When rolling a six-sided die, what is the probability of rolling a number greater than 1 or an even number?

• Numbers greater than 1: 2, 3, 4, 5, 6 (5 outcomes)

• Even numbers: 2, 4, 6 (3 outcomes)

• Overlap: 2, 4, 6 (3 outcomes)

• $P(>1) = \frac{5}{6}$, $P(\text{even}) = \frac{3}{6}$, $P(>1 \cap \text{even}) = \frac{3}{6}$

• $P(>1 \cup \text{even}) = \frac{5}{6} + \frac{3}{6} - \frac{3}{6} = \frac{5}{6} = 0.83$

Using Tables to Find Probabilities

Tables can be used to organize outcomes and calculate probabilities for events, especially when dealing with categorical data.

	Wearing a red shirt	Wearing a green shirt	Total
Wearing Pants	48	47	112
Wearing Shorts	63	125	188
Total	111	172	300

• Example: What is the probability that a randomly selected person is wearing shorts or a green shirt?

• Wearing shorts: 188

• Wearing green shirt: 172

• Wearing shorts and green shirt: 125

• $P(\text{shorts or green shirt}) = \frac{188}{300} + \frac{172}{300} - \frac{125}{300} = \frac{235}{300} = 0.78$

Practice Problems

• Mutually Exclusive: For two mutually exclusive events A and B, compute $P(A \cup B)$ if $P(A) = 0.15$ and $P(B) = 0.32$.

• Solution: $P(A \cup B) = 0.15 + 0.32 = 0.47$

• Non-Mutually Exclusive: A card is drawn from a standard deck of 52 cards. What is the probability that the card is a diamond or a king?

• Diamonds: 13, Kings: 4, King of diamonds: 1 (overlap)

• $P(\text{diamond or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = 0.31$

Additional info: The notes cover the addition rule for probability, including both mutually exclusive and non-mutually exclusive events, with examples and applications relevant to Chapter 5: Probability in Our Daily Lives.