SECTION 3.2 Conditional Probability and the Multiplication Rule 149 The Multiplication Rule To find the probability of two events occurring in sequence, you can use the Multiplication Rule. The probability that two events A and B will occur in sequence is P1A and B2 = P1A2 # P1B0 A2. Events A and B are dependent. If events A and B are independent, then the rule can be simplified to P1A and B2 = P1A2 # P1B2. Events A and B are independent. This simplified rule can be extended to any number of independent events. The Multiplication Rule for the Probability of A and B Using the Multiplication Rule to Find Probabilities 1. Two cards are selected, without replacing the first card, from a standard deck of 52 playing cards. Find the probability of selecting a king and then selecting a queen. 2. A coin is tossed and a die is rolled. Find the probability of tossing a head and then rolling a 6. SOLUTION 1. Because the first card is not replaced, the events are dependent. P1K and Q2 = P1K2 # P1Q0 K2 = 4 52 # 4 51 = 16 2652 ≈ 0.006 So, the probability of selecting a king and then a queen without replacement is about 0.006. 2. The events are independent. P1H and 62 = P1H2 # P162 = 1 2 # 1 6 = 1 12 ≈ 0.083 So, the probability of tossing a head and then rolling a 6 is about 0.083. TRY IT YOURSELF 3 1. The probability that a salmon swims successfully through a dam is 0.85. Find the probability that two salmon swim successfully through the dam. 2. Two cards are selected from a standard deck of 52 playing cards without replacement. Find the probability that they are both hearts. Answer: Page A38 EXAMPLE 3 Study Tip In words, to use the Multiplication Rule, 1. find the probability that the first event occurs, 2. find the probability that the second event occurs given that the first event has occurred, and 3. multiply these two probabilities. Study Tip Recall from Section 3.1 that a probability of 0.05 or less is typically considered unusual. In the first part of Example 3, 0.006 6 0.05. This means that selecting a king and then a queen (without replacement) from a standard deck is an unusual event.
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