Conditional Probability and General Multiplication Rule AP Statistics

Notice how conditional probability “shrinks the sample space”. Ex. 1) What is the probability of drawing a 7 of clubs, then a 4 of hearts without replacement? P(draw 7 clubs) P(draw a 4 of hearts 7 of clubs was drawn) = Notice how conditional probability “shrinks the sample space”. “given that”

Conditional Probabilities Here is a contingency table that gives the counts of AP Stats students by their gender and political views. (Data are from Fall 2005 Class Survey) P(Female) = 77/137 = 0.562 P(Female and Liberal) = 30/137 = 0.219 What is the probability that a selected student has moderate political views given that we have selected a female?

Conditional Probabilities (continued) What is the probability that a selected student has moderate political views given that we have selected a female? P(Moderate | Female) = 24/77 = 0.311 Conditional probability, P (B|A) – the probability of event B given event A.

Conditional Probabilities (continued) Formal Definition: Example: P(Moderate and Female) P(Female) =(24/137) / (77/137) = 0.175 / 0.562 = 0.311

Multiplication Rule for Independent Events Revisited P(A and B) = P(A) * P(B) Independent – the occurrence of one event has no effect on the probability of the occurrence of another event. Example: A survey by the American Automobile Association (AAA) revealed that 60 percent of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year? P(R1 and R2) = P(R1)*P(R2) = (0.6)*(0.6) = .36

General Multiplication Rule Use when events are Dependent. P(A and B) = P(A) * P(B|A) For two events A and B, the joint probability that both events will happen is found by multiplying the probability event A will happen by the conditional probability of event B occurring.

General Multiplication Rule **Also:

When events are independent:

when events A and B are independent! When not independent:

Ex 1) In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is .10. If the student is female, what is the probability that she plays sports?

Ex 2) The probability that a randomly selected student plays sports if (given that) they are male is .31. What is the probability that the student is male and plays sports if the probability that they are male is .49?

Checking for Independence The probability of a student in this course getting an A is 0.35; the probability of being female is 0.60. If P(A and Female) = 0.15, are the events “getting an A” and “being a female” independent? Check to see if P(A)•P(Female) = P(A and Female) (0.35)(0.60) ≠0.15 Thus, the events “getting and A” and “being a female” are not independent.

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