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Conditional Probability and the Multiplication Rule Unit 3.3

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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.

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Multiplication Rule Tip Find the probability the first event occurs. Find the probability the second event occurs given the first event has occurred and Multiply these two probabilities.

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Using the Multiplication Rule to find Probability A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.

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Using the Multiplication Rule to find Probability 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.

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Using the Multiplication Rule to find Probability The probability that a salmon swims successfully through a dam is 0.85. Find the probability that two salmon successfully swim through the dam.

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Using the Multiplication Rule to find Probability Two cards are selected from a standard deck without replacement. Find the probability that both are hearts.

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Using the Multiplication Rule to find Probability 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.

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Using the Multiplication Rule to find Probability The probability that a salmon swims successfully through a dam is 0.85. Find the probability that three salmon swim successfully through the dam.

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Using the Multiplication Rule to find Probability Find the probability that none of the three salmon are successful.

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Using the Multiplication Rule to find Probability Find the probability that at least one of the three salmon is successful in swimming through the dam.

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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

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Conditional Probability To find probabilities when events are dependent, use the multiplication rule with a modification in notation. P(A and B) = P (A) ● P (B\A).

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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

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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

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Finding Conditional Probability 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

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Finding Conditional Probability Find the probability that a child does not have the gene. Find the probability that a child does not have the gene, given that the child has a normal IQ.