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# 5.4 - STAT3000 Section 5.4 Conditional Probability Suppose...

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STAT3000 Section 5.4: Conditional Probability Suppose you draw 2 cards (with replacement) from a normal deck of cards. What is the probability you will draw one spade and then a heart? A = first card is a spade, B = second card is a heart P(A and B) = What if these drawings are done without replacement? P(A and B) = Ex : The probability of observing an even number (event A) on a toss of a fair die is 0.5, where S = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. Suppose we’re given the information that on a particular throw of the die the result was a number less than or equal to 3 (event B), so B = {1, 2, 3}. Would the probability of observing an even number on that throw of the die still be equal to 0.5? 66

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Ex : A federal agency concerned with rising health costs and employees without health care, gathered the following information from several randomly selected companies nationwide within the same industry. Company Size Fee-for-Service PPO HMO Totals Small 1808 1757 1456 5021 Medium 8953 6491 6983 22,382 Large 330,419 241,77 0 233,71 1 805,900 Totals 341,180 250,01 8 242,10 5 833,303 One employee from the 833,303 total employees is to be chosen at random for further analysis. A: an employee who chose fee-for-service B: an employee from a small company *Over lap so there cannot be disjoint a. Find P(B)=5021/833303=.006025 b. Find P(A and B)= 1808/833303=.00217 c. Find P(A or B)= 341180/833303+5021/833303- 1808/833303=.4132 d. Find P(A | B)= P(A given that B has occurred) =P( fee for service given that he works for small company)=1808/5021= 67
Numerator= P(A and B) Denominator= P(B) e. Find P(B | A) P( small company , given that fee-for- service)=1808/341180 Numerator= P(A and B) Denominator= P(A) Conditional Probability Strategies 1. The Formula To find the conditional probability that event A occurs given that event B occurs, divide the probability that both A and B occur by the probability that B occurs. P(A | B) = P(A and B) ______ _____________ P(B)_____________ 68

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2. Tree Diagram Revisit the dice example, but use a tree diagram. P(A | B) = __________ * __________ = If knowing that B occurs gives no additional information about the probability of A occurring, then A and B are independent events. If P(A | B) = P(A) or P(B | A) = P(B), then A and B are independent.
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