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Stat104-Section3_sol

Stat104-Section3_sol - Stat 104 Section 3 Handout Solutions...

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Stat 104, Section 3 Handout Solutions Benny, Thursday, 4:00pm, SC 309 Key concepts Probability and Conditional Probability Unions, intersections, complements, disjoint events 2x2 tables Independence of events Bayes’ Rule/Theorem Practice Problems 1. M&M’s - sample space, basic probability and cond. prob. Playing with sets. I have 5 M&M’s in a bag. 2 are blue, 2 are red, and 1 is yellow. I plan on eating 2 more. a. Write the sample space of outcomes for pairs of M&M’s that I will eat? (Here and afterwards, order does not matter) S = { ( b 1 , b 2 ) , ( b 1 , r 2 ) , ( b 2 , r 1 ) , ( b 1 , r 2 ) , ( b 2 , r 2 ) , ( b 1 , y ) , ( b 2 , y ) , ( r 1 , y ) , ( r 2 , y ) , ( r 1 , r 2 ) } Note that we label M&M’s with the same color separately to account for the fact that they are more frequently found. b. Write down the set of events for: A: eating two mismatched M&M’s (different colors) A = { ( b 1 , r 1 ) , ( b 2 , r 1 ) , ( b 1 , r 2 ) , ( b 2 , r 2 ) , ( b 1 , y ) , ( b 2 , y ) , ( r 1 , y ) , ( r 2 , y ) } B: eating at least one blue M&M B = { ( b 1 , b 2 ) , ( b 1 , r 1 ) , ( b 2 , r 1 ) , ( b 1 , r 2 ) , ( b 2 , r 2 ) , ( b 1 , y ) , ( b 2 , y ) } c. What is the probability of A and B? P ( A ) = # outcomes in A # of outcomes in S = | A | | S | = . 8 P ( B ) = # outcomes in B # of outcomes in S = | B | | S | = . 7 1
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d. Are A & B disjoint? How do you know? No for several reasons. In terms of their set representation, there are outcomes that appear in both A and B. Numerically, P ( A ) + P ( B ) > 1, which is not possible for disjoint events. Finally, it is intuitively clear that just because one eats two mismatched M&M’s (A) does not mean one cannot eat at least one blue M&M.
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