Math121 Discussion - 13th week solutions

Math121 Discussion - 13th week solutions - MATH 121...

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MATH 121 DISCUSSION WEEK 13 SOLUTIONS 1) (a) Call our random variable W where W represents # of white socks drawn. Note you are drawing two socks without replacement. So: P(W=0) = P(B1 B2) = P(B1) × P(B2 | B1) = ( 5 8 )( 4 7 ) = 20 56 P(W=1) = P(B1 W2) + P(W1 B2) = P(B1) × P(W2 | B1) + P(W1) × P(B2 | W1) = ( 5 8 )( 3 7 ) + ( 3 8 )( 5 7 ) = 30 56 P(W=2) = P(W1 W2) = P(W1) × P(W2 | W1) = ( 3 8 )( 2 7 ) = 6 56 Table 1. Probability Distribution for W W Outcomes Probability 0 (B1, B2) 20 56 1 (B1, W2), (W1, B2) 30 56 2 (W1, W2) 6 56 (b) See separate attachment for probability distribution plot. (c) E ( W ) = 0 × 20 56 + 1 × 30 56 + 2 × 6 56 = 42 56 = 0 . 75. If you were to do this repeatedly, your long-run average of white socks drawn would be 0.75. 2) (a) Let W = net change in earnings. It is very important to note that when you are betting money, you do not actually lose it until after you’ve lost the bet. This is diﬀerent from paying for something, in which case, you must deduct this value in all your outcomes. Table 2. Probability Distribution for W W i Outcomes Probability W 1 = 50 Lands on green 2 38 W 2 = - 5 Lands on red or black 36 38 (b) See separate attachment for probability distribution plot. (c) E ( W ) = W 1 × P ( W 1 ) + W 2 × P ( W 2 ) = 50 × 2 38 + ( - 5) × 36 38 = - 80 38 ≈ - \$2 . 10 . No, you should not make repeated bets because you should expect to lose money on it. (d) In this case, we would need to ﬁnd a diﬀerent W 1 so that E ( W ) > 0. We can solve for such a W 1 by setting E(W) = 0. 0 = W 1 × 2 38 + ( - 5) × 36 38 . W 1 × 2 38 = 180 38 . W 1 = 180 38 × 38 2 = \$90 . So there is no value gained or lost if the reward is \$90.00. In order for it to be worth making repeated bets, the payout should be greater than \$90.00 . If that is the case, then E ( W ) > 0, so it is worth playing multiple times in that case. 1

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2 MATH 121 DISCUSSION WEEK 13 SOLUTIONS 3) (a) If you are not familiar with roulette: there are 38 total compartments on a roulette wheel. There are 2 green compartments numbered 0 and 00 and 18 red and 18 black compartments numbered 1 through 36. You win prizes if the ball lands in the compartment you bet on while the wheel is spinning. Based on the information given: P(win \$50) = 2 38 . The possible compartments are 0 and 00. P(win \$10) =
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This note was uploaded on 12/07/2010 for the course MATH Math 121 taught by Professor Beaulieu during the Fall '10 term at UMass (Amherst).

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Math121 Discussion - 13th week solutions - MATH 121...

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