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Conditional Probability
5.4 Conditional Probability
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Recall For “or” probabilities For “and” probabilities
The Addition Rule applies to two disjoint events … the “easy” case The General Addition Rule applies to any two events For “and” probabilities The Multiplication Rule applies to two independent events … the “easy” case The General Multiplication Rule, this section, applies to any two events
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Example Example Choosing cards from a deck of cards E = we chose a diamond as the first card We did not replace our first card F = we chose a heart as the second card The probability of F happening, given that E has already happened, is 13/51 There are 51 cards remaining 13 of them are hearts
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Conditional Probability
13/51 is called a conditional probability The probability of choosing a heart is 13/52 The probability of choosing a heart, given that we had already chosen a diamond, is 13/51 This can be written P(Diamond | Heart) = 13/51
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Conditional Probability
The notation for conditional probability P(F|E) is the probability of F given event E Only the outcomes contained in the event E are included in computing conditional probabilities
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Example A group of adults are as per the following table
We choose a person at random out of this group If E = “male” and F = “left handed”, compute P(F) and P(F|E) Male Female Total Right Handed 38 42 80 Left Handed 12 8 20 50 100
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If E = “male” and F = “left handed”, compute P(F) and P(F|E)
F = “left handed” … P(F) = 20/100 = 0. 20 E = “male” … P(F|E) = probability of left handed, given male = 12/50 = 0.24 There are 50 males and 12 of them are left handed The probability of left handed, given male, is 12/50
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Conditional Probability Rule
The Conditional Probability Rule is An interpretation of this is that we only consider the cases when E occurs (i.e. P(E)), and out of those, we consider the cases when F occurs (i.e. P(E and F), since E always has to occur)
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General Multiplication Rule
We can take the Conditional Probability Rule and rearrange it to be This is the General Multiplication Rule
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Example Example For a student in a statistics class
E = “did not do the homework” with P(E) = 0.2 F = “the professor asks that student a question about the homework” with P(F|E) = .9 What is the probability that the student did not do the homework and the professor asks that student a question about the homework? P(E and F) = P(E) • P(F|E) = 0.2 • 0.9 = 0.18
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Summary Conditional probabilities P(F|E) represent the chance that F occurs, given that E occurs also The General Multiplication Rule applies to “and” problems for all events and involves conditional probabilities
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Examples Suppose a single card is selected from a standard 52- card deck. What is the probability that the card drawn is a king? Now suppose a single card is drawn from a standard 52- card deck, but we are told that the card is a heart. What is the probability that the card drawn is a king?
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Example According to the U. S. National Center for Health Statistics, in 2002, 0.2% of deaths in the United States were 25- to 34- year- olds whose cause of death was cancer. In addition, 1.97% of all those who died were 25 to 34 years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been 25 to 34 years old?
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Example According to the U. S Census Bureau, 19.1% of U. S. households are in the Northeast. In addition, 4.4% of U. S. households earn $75,000 per year or more and are located in the Northeast. Determine the probability that a randomly selected U. S. household earns more than $ 75,000 per year, given that the household is located in the Northeast.
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Example Died from Cancer Did not die from Cancer Never Smoked Cigars 782 120474 Former Cigar Smoker 91 7757 Current Cigar Smoker 141 7725 ( a) What is the probability that a randomly selected individual from the study who died from cancer was a former cigar smoker?
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Example ( b) What is the probability that a randomly selected individual from the study who was a former cigar smoker died from cancer?
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Example A bag of 30 tulip bulbs purchased from a nursery contains 12 red tulip bulbs, 10 yellow tulip bulbs, and 8 purple tulip bulbs. ( a) What is the probability that two randomly selected tulip bulbs are both red? ( b) What is the probability that the first bulb selected is red and the second yellow? ( c) What is the probability that the first bulb selected is yellow and the second is red?
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Example Due to a manufacturing error, three cans of regular soda were accidentally filled with diet soda and placed into a 12- pack. Suppose that two cans are randomly selected from the case. ( a) Determine the probability that both contain diet soda. ( b) Determine the probability that both contain regular soda. Would this be unusual?
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