Review Problems Solutions

# Review Problems Solutions - Test 2/Chapter 8 Review Problem...

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Test 2/Chapter 8 Review Problem 1 An unbiased coin is tossed 7 times. What is the probability that (A) heads is tossed exactly 5 times? Solution: n ( E ) = C (7 , 5) n ( S ) = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 7 P ( E ) = C (7 , 5) 2 7 . 1641 (B) tails is tossed exactly 3 times? Solution: n ( E ) = C (7 , 3) n ( S ) = 2 7 P ( E ) = C (7 , 3) 2 7 . 2734 (C) heads is tossed on the first and third toss? Solution: n ( E ) = 1 × 2 × 1 × 2 × 2 × 2 × 2 = 2 5 n ( S ) = 2 7 P ( E ) = 2 5 2 7 = . 25 (D) heads is tossed on the second toss and tails is tossed on the fourth toss? Solution: n ( E ) = 2 × 1 × 2 × 1 × 2 × 2 × 2 = 2 5 n ( S ) = 2 7 P ( E ) = 2 5 2 7 = . 25 (E) heads is tossed at least once? Solution: Let E be the event that heads is tossed at least once. Then E c is the event that heads is not tossed. n ( E c ) = 1 × 1 × 1 × 1 × 1 × 1 = 1 since tails must be tossed on each toss. n ( S ) = 2 7 P ( E c ) = 1 2 7 P ( E ) = 1 P ( E c ) = 1 1 2 7 . 0078 (F) tails is tossed at least five times? Solution: Tails is tossed at least five times implies that tails can be tossed 5 times OR 6 times OR 7 times. n ( E ) = C (7 , 5) + C (7 , 6) + C (7 , 7) n ( S ) = 2 7 P ( E ) = C (7 , 5)+ C (7 , 6)+ C (7 , 7) 2 7 . 2266 1

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Problem 2 Four cards are selected at random without replacement from a well-shu ffl ed deck of 52 playing cards. Find the probability that (A) all of the cards are hearts? Solution: There are 13 hearts and we want 4 of them. n ( E ) = C (13 , 4) n ( S ) = C (52 , 4) P ( E ) = C (13 , 4) C (52 , 4) . 0026 (B) exactly 2 of the cards are diamonds? Solution: There are 13 diamonds and we want 2 of them. There are 39 non-diamonds and we want 2 of them (since we need a total of 4 cards). n ( E ) = C (13 , 2) C (39 , 2) n ( S ) = C (52 , 4) P ( E ) = C (13 , 2) C (39 , 2) C (52 , 4) . 2135 (C) two of the cards are face cards (i.e. king, queen, jack) and the other two cars are fours? Solution: There are 12 face cards and we want 2 of them. There are 4 fours and we want two of them. n ( E ) = C (12 , 2) C (4 , 2) n ( S ) = C (52 , 4) P ( E ) = C (12 , 2) C (4 , 2) C (52 , 4) . 0015 (D) none of the cards are red cards? Solution: There are 26 red cards and we want 0 of them. There are 26 black cards and we want 4 of them. n ( E ) = C (26 , 0) C (26 , 4) n ( S ) = C (54 , 4) P ( E ) = C (26 , 0) C (26 , 4) C (52 , 4) . 0552 (E) one of the cards is a club, two of the cards are spades, and the other card is a diamond? Solution: There are 13 clubs and we want 1 of them. There are 13 spades and we want 2 of them. There are 13 diamonds and we want 1 of them. n ( E ) = C (13 , 1) C (13 , 2) C (13 , 1) n ( S ) = C (52 , 4) P ( E ) = C (13 , 1) C (13 , 2) C (13 , 1) C (52 , 4) . 0487 2
Problem 3 A quiz consists of 5 true-or-false questions. If a student guesses at every answer, what is the probability that he or she will answer (A) exactly 3 of the questions correctly?

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