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1. A coin is weighted so that heads is twice as likely to appear as tails. Find P(T) and P(H). 1 2 ANS: P(T) = 3 ; P(H) = 3 . 2. Two men, m1 and m2, and three women, w1, w2, w3, are in a chess tournament. Those of the same sex have equal probabilities of winning, but each man is twice as likely to win as any women. (i) Find the probability that a woman wins the tournament. (ii) If m1 and 3 w1 are married, find the probability that one of them wins the tournament. ANS: (i) 7 3 (ii) 7 . 3. A coin is weighted so that heads is three times as likely to appear as tails. Find P(T) and 3 1 P(H). ANS: P(H) = 4 ; P(T) = 4 . 4. Three students A, B and C are in a swimming race. A and B have the same probability of winning and each is twice as likely to win as C. Find the probability that B or C wins. 3 ANS: 5 5. A die is weighted so that the even numbers have the same chance of appearing, the odd numbers have the same chance of appearing, and each even number is twice as likely to appear as any odd number. Find the probability that (i) an even number appears, (ii) a prime number appears, (iii) an odd number appears, (iv) an odd prime number appears. 2 4 1 2 ANS: (i) 3 ; (ii) 9 ; (iii) 3 ; (iv) 9 . 6. Which function defines a probability space on S = {a1, a2, a3}? 1 1 1 (i) P(a1) = 4 , P(a2) = 3 , P(a3) = 2 . 2 (ii) P(a1) = 3 , 1 P(a2) = 3 , 2 P(a3) = 3 . 1 (iii) P(a1) = 6 , 1 P(a2) = 3 , 1 P(a3) = 2 . (iv) P(a1) = 0, ANS: (i) No 1 P(a2) = 3 , (ii) No 2 P(a3) = 3 . (iii) Yes (iv) Yes 1 7. Let P be a probability function on S = {a1, a2, a3}. Find P(a1) if (i) P(a2) = 3 and P(a3) = 1 1 4 ; (ii) P(a1) = 2P(a2) and P(a3) = 4 ; (iii) P(a2 or a3) = 2P(a1); (iv) P(a3) = 2P(a2) and P(a2) = 3P(a1) 5 ANS: (i) 12 ; 1 (ii) 2 ; 1 (iii) 3 ; 1 (iv) 10 . 8. A class contains 10 men and 20 women of which half the men and half the women have brown eyes. Find the probability that a person chosen at random is a man or has brown 2 eyes. ANS: 3 9. A class contains 5 freshmen, 4 sophomores, 8 juniors and 3 seniors. A student is chosen at random to represent the class. Find the probability that the student is (i) a sophomore, 1 3 11 (ii) a senior, (iii) a junior or senior. ANS: (i) 5 ; (ii) 20 ; (iii) 20 .