Compare “A and B” to “A or B” The compound event “A and B” means that A and B both occur in the same trial. Use the multiplication rule to find P(A and B). The compound event “A or B” means either A can occur without B, B can occur without A or both A and B can occur. Use the addition rule to find P(A or B). A B A B A or B A and B

Mutually Exclusive Events Two events, A and B, are mutually exclusive if they cannot occur in the same trial. A = A person is under 21 years old B = A person is running for the U.S. Senate A = A person was born in Philadelphia B = A person was born in Houston A Mutually exclusive B P(A and B) = 0 When event A occurs it excludes event B in the same trial.

Non-Mutually Exclusive Events If two events can occur in the same trial, they are non-mutually exclusive. A = A person is under 25 years old B = A person is a lawyer A = A person was born in Philadelphia B = A person watches West Wing on TV A and B Non-mutually exclusive P(A and B) ≠ 0 A B

The Addition Rule The probability that one or the other of two events will occur is: P(A) + P(B) – P(A and B) A card is drawn from a deck. Find the probability it is a king or it is red. A = the card is a king B = the card is red. P(A) = 4/52 and P(B) = 26/52 but P(A and B) = 2/52 P(A or B) = 4/52 + 26/52 – 2/52 = 28/52 = 0.538

When events are mutually exclusive, The Addition Rule A card is drawn from a deck. Find the probability the card is a king or a 10. A = the card is a king B = the card is a 10. P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52 P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054 When events are mutually exclusive, P(A or B) = P(A) + P(B)

Contingency Table One of the responses is selected at random. Find: The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 1. P(Miami and Yes) 2. P(Miami and Seattle) 3. P(Miami or Yes) 4. P(Miami or Seattle)

Contingency Table 1. P(Miami and Yes) 2. P(Miami and Seattle) = 0 Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 1. P(Miami and Yes) 2. P(Miami and Seattle) = 250/1000 • 150/250 = 150/1000 = 0.15 = 0

Contingency Table 3 P(Miami or Yes) 4. P(Miami or Seattle) Omaha Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 3 P(Miami or Yes) 4. P(Miami or Seattle) 250/1000 + 400/1000 – 150/1000 = 500/1000 = 0.5 250/1000 + 450/1000 – 0/1000 = 700/1000 = 0.7

P(A and B) = P(A) • P(B|A) P(A or B) = P(A) + P(B) - P(A and B) Summary For complementary events P(E') = 1 - P(E) Subtract the probability of the event from one. The probability both of two events occur P(A and B) = P(A) • P(B|A) Multiply the probability of the first event by the conditional probability the second event occurs, given the first occurred. Probability at least one of two events occur P(A or B) = P(A) + P(B) - P(A and B) Add the simple probabilities, but to prevent double counting, don’t forget to subtract the probability of both occurring.