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5.4

5.4. Conditional Probability. Recall. For “or” 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

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5.4

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  1. 5.4 Conditional Probability

  2. Recall • For “or” 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

  3. 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

  4. Conditional Probability • 13/51 is called a conditional probability • The probability of choosing a heart is 13/52 • The probability of choosing a heart, giventhat we had already chosen a diamond, is 13/51 • This can be written P(Diamond | Heart) = 13/51

  5. Conditional Probability • The notation for conditionalprobability 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

  6. 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)

  7. 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

  8. Conditional Probability Rule • The ConditionalProbabilityRule 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)

  9. General Multiplication Rule • We can take the Conditional Probability Rule and rearrange it to be • This is the GeneralMultiplicationRule

  10. 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

  11. 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

  12. 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?

  13. 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?

  14. 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.

  15. Example ( a) What is the probability that a randomly selected individual from the study who died from cancer was a former cigar smoker?

  16. Example ( b) What is the probability that a randomly selected individual from the study who was a former cigar smoker died from cancer?

  17. 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?

  18. 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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