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Unit 3.3 CONDITIONAL PROBABILITY AND THE MULTIPLICATION RULE Multiplication Rule The Multiplication Rule can be used to find the probability of two or more events that occur in a sequence . The multiplication Rule for the probability of A and B If events A and B are independent, then the rule can be simplified to P(A and B) = P (A) ● P (B). This simplified rule can be extended for any number of independent events. Multiplication Rule Tip 1. Find the probability the first event occurs. 2. Find the probability the second event occurs given the first event has occurred and 3. Multiply these two probabilities. Using the Multiplication Rule to find Probability 1. A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6. Using the Multiplication Rule to find Probability 1. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of selecting a Ace and then selecting a queen. Using the Multiplication Rule to find Probability 1. The probability that a salmon swims successfully through a dam is 0.85. Find the probability that two salmon successfully swim through the dam. Using the Multiplication Rule to find Probability 1. Two cards are selected from a standard deck without replacement. Find the probability that both are hearts. Using the Multiplication Rule to find Probability 1. A Harris poll found the 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say they suffer great stress at least once a week. Using the Multiplication Rule to find Probability 1. The probability that a salmon swims successfully through a dam is 0.85. Find the probability that three salmon swim successfully through the dam. Using the Multiplication Rule to find Probability 1. Find the probability that none of the three salmon are successful. Using the Multiplication Rule to find Probability 1. Find the probability that at least one of the three salmon is successful in swimming through the dam. Dependent Events When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events. Examples Drawing a card from a deck, NOT replacing it, and then drawing a second card. Being a lifeguard and getting a tan. Having high grades and getting a scholarship Conditional Probability To find probabilities when events are dependent, use the multiplication rule with a The probability of B modification in notation. given that event A P(A and B) = P (A) ● P (B\A). has already occured Finding Conditional Probability Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced) Solution: Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens. So, P(B|A) = 4/51 = 0.078 Finding Conditional Probability Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events. Getting 3 Jacks Getting an ace, a king, and a queen in order Getting a club, a spade, and a heart in order Getting three clubs Finding Conditional Probability Gene Present Gene Not Present Total High IQ 33 19 52 Normal IQ 39 11 50 Total 72 30 102 The table at the left shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene. Solution: There are 72 children who have the gene. So, the sample space consists of these 72 children, as shown at the left. Of theses, 33 have a high IQ. So, P(B\A)= 33/72 =0.458 Finding Conditional Probability Gene Present Gene Not Present Total High IQ 33 19 52 Normal IQ 39 11 50 Total 72 30 102 1. Find the probability that a child does not have the gene. 2. Find the probability that a child does not have the gene, given that the child has a normal IQ.