4-2 Addition Rule and Multiplication Rule 161 Multiplication Rule Notation for Multiplication Rule We begin with basic notation followed by the multiplication rule. We strongly suggest using the intuitive multiplication rule, because it is based on understanding instead of blind use of a formula. Notation P1A and B2 = P1event A occurs in a first trial and event B occurs in a second trial2 P1B A2 represents the probability of event B occurring after it is assumed that event A has already occurred. (Interpret B A as “event B occurs after event A has already occurred.”) SOLUTION Using the rule of complementary events, we get P1household not having a smartphone2 = 1 - P1household with a smartphone2 = 1 - 0.87 = 0.13 The probability of randomly selecting a household not having a smartphone is 0.13. YOUR TURN. Do Exercise 7 “Laundry Symbols.” CAUTION The notation P1A and B2 has two meanings, depending on its context. For the multiplication rule, P1A and B2 denotes that event A occurs in one trial and event B occurs in another trial; for the addition rule we use P1A and B2 to denote that events A and B both occur in the same trial. INTUITIVE MULTIPLICATION RULE To find the probability that event A occurs in one trial and event B occurs in another trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B is found by assuming that event A has already occurred. FORMAL MULTIPLICATION RULE P1A and B2 = P1A2 # P1B A2 Independence and the Multiplication Rule When applying the multiplication rule and considering whether the probability of event B must be adjusted to account for the previous occurrence of event A, we are focusing on whether events A and B are independent. DEFINITIONS Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are independent if the occurrence of any does not affect the probabilities of the occurrence of the others.) If A and B are not independent, they are said to be dependent.
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