PROBABILITY: Combining Two Events Jonathan Fei

Definitions RANDOM EXPERIMENT: any procedure or situation that produces a definite outcome that may not be predictable in advance OUTCOME: a single possible result from a random experiment EVENT: Any collection of outcomes Siegel, Andrew F. and Charles J Morgan, Statistics and Data Analysis: An introduction John Wiley & Sons, Inc. 1996

Example RANDOM EXPERIMENTS Flipping a coin 5 times and counting the number of heads OUTCOME You get 2 Heads and 3 Tails EVENT Getting more Heads than Tails

OR When one, the other, or both events are true If Dave has a red shirt OR a green hat, then: He has a red shirt He has a green hat He has a red shirt and a green hat

Mutually Exclusive When two events cannot occur at the same time Same as disjoint When you flip a coin, getting heads and getting tails are mutually exclusive. You either get heads or get tails.

AND When two events are both true If Dave has a red shirt AND a green hat, then: He has a red shirt and a green hat

Formulas P(A or B) = P(A) + P(B) – P(A and B) P(A and B) = P(A) + P(B) – P(A or B) If mutually exclusive, P(A or B) = P(A) + P(B)

Conditional Probabilities For P(A given B) or P(A | B): It is the probability of A if you know the probability of B P(A | B) = P(A and B) / P(B) Example: In a standard 52 card deck, if you choose two cards and you know the first card was red, it changes the probability that the second card will be black.

Independent Events Two events are independent if information about one does not affect the other If something is independent, then: P(A and B) = P(A) P(B) Example: If you flip a coin and get heads, it does not affect the chance of getting heads or tails on the next flip. The probability is the same.

IMPORTANT DISTINCTIONS Mutually Exclusive and Independence are NOT the same thing Something mutually exclusive cannot be independent

Example For example, the events A = being a senior in high school and B = being a freshman are mutually exclusive, since if I know that someone is a senior, they cannot be a freshman. In order for these two events to be independent, knowing that someone is a senior would have to not give me any information about whether a student is a freshman or not, which is obviously not the case. Todd Frost Flintridge Preparatory School