Practice Midterm 2 WI 10 Solutions

# Note that having a green belt is independent of

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Note that having a green belt is independent of gender because P(G|F) = P(G|M) = 0.80. So you can just say that P(G) = 0.80 by definition of independence. P(G) = P(G and F) + P(G and M) = P(G|F) P(F) + P(G|M) P(M) = (0.8)(0.3) + (0.8)(0.7) = 0.8 ANS__ 0.8 _____ 4

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22. What percentage of the green belts in Tao’s class are male? Note that having a green belt is independent of gender because P(G|F) = P(G|M) = 0.80. So you can just say that P(M) = 0.70, which is given in the problem. 7 . 0 8 . 0 ) 7 . 0 )( 8 . 0 ( ) ( ) ( ) | ( ) ( ) ( ) | ( = = = = G P M P M G P G P G and M P G M P ANS___ 0.7 ____ 23. Using this information fill in a two-way table of all the probabilities for this situation (including all marginals). P(G and M) = P(G|M) P(M) = (0.8) (0.7) = 0.56 P(G and F) = P(G|F) P(F) = (0.8) (0.3) = 0.24 G G C M 0.56 0.14 0.7 F 0.24 0.06 0.3 0.8 0.2 1 5
Problems 24-26. Suppose the percentage of children in Tao’s karate class is 10% and the percentage of adults is 90%. Also suppose that 30% of the children are new to the class and 20 percent of the adults are new to the class. P(C) = 0.1 P(A) = 0.9 P(N|C) = 0.3 P(N|A) = 0.2 24. Using this information fill in a two-way table of all the probabilities (including all marginals). P(N and C) = P(N|C) P(C) = (0.3) (0.1) = 0.03 P(N and A) = P(N|A) P(A) = (0.2) (0.9) = 0.18 N N C C 0.03 0.07 0.1 A 0.18 0.72 0.9 0.21 0.79 1 25. What is the probability that someone in Tao’s karate class is new? (You can answer this with or without the table in #24.) P(N) = P(N and C) + P(N and A) = P(N|C) P(C) + P(N|A) P(A) = (0.3) (0.1) + (0.2) (0.9) = 0.03 + 0.18 = 0.21 ANS__ 0.21 ___ 26. What percentage of the new class members are children? (You can answer this with or without the table in #24.) 143 . 0 21 . 0 03 . 0 21 . 0 ) 1 . 0 )( 3 . 0 ( ) ( ) ( ) | ( ) ( ) ( ) | ( = = = = = N P C P C N P N P N and C P N C P ANS__ 0.143 ___ 6

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Problems 27-30. The exam scores in Mr. Math’s class were: 40% A’s, 35% B’s, 25% C’s, No D’s and No E’s. Select two of his students at random and record their grades. P(A) = 0.4 P(B) = 0.35 P(C) = 0.25 27. Write down the sample space for this problem. S = {AA, AB, AC, BA, BB, BC, CA, CB, CC} 28. What is the chance that both people got the same grade? P(Both Same Grade) = P(AA or BB or CC) = P(AA) + P(BB) + P(CC) = P(A) P(A) + P(B) P(B) + P(C) P(C) = (0.4) (0.4) + (0.35) (0.35) + (0.25) (0.25) = 0.345 ANS__ 0.345 ____ 29. What is the chance that at least one of them got an A?
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