Unit 7 FRQ

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Auburn University *

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2010

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Industrial Engineering

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May 15, 2024

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AP Statistics Unit 7 Random Variables Free Response Directions: Complete the assignment on this paper. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit. 1. A Roulette wheel has 38 slots numbered 0 to 36 and 00. The wheel is spun and a ball is thrown into the wheel and comes to rest in one of the slots. There are numerous of ways to bet, individual numbers, groups of numbers (1-12, 13-24, etc.), by color (half of the numbers are black and the half are red), and in various other combinations. This problem is going to focus on betting $1.00 on the number group 1-12. If the ball lands in any of the values 1-12 the bet is won and the return is $3.00. If the ball lands on any of the other values the bet is lost. a.) Compute the expected value of winnings for this game. (4 points) Event X Frequency P(x) X*P(x) Win $2.00 12 0.3158 0.6316 Lose -$1.00 26 0.6842 -0.6842 0.0526 b.) Interpret this expected value. (4 points) The average player will lose an average of 5 cents per spin of the Roulette wheel, meaning that the casino will earn an average of 5 cents per round. c.) What is the average return to the casino from 1,000,000 such bets? (4 points) The casino will earn around $50,000 from 1,000,000 bets since they make 5 cents per round. 2. A venture capital fund has the mandate to invest in new businesses that may be perceived as being too risky by other investors. Suppose that a fund invests its funds in units with no more than one unit per firm in order to diversify its investments. Furthermore, suppose there are two classes of firms with the following distributions of net returns, and assume that all firms operate independently of each other. Type A firms Type B firms % returns -15 0 45 -25 -5 0 25 65 Probability .30 .30 .40 .15 .15 .30 .20 .20 a.) Both firms have the same expected return. Compute and interpret it. (6 points)
Type A Firms % Returns X P(x) X*P(x) -15 -0.15 0.3 -0.045 0 0 0.3 0 45 0.45 0.4 0.18 Expected return 0.135 Type B Firms % Returns X P(x) X*P(x) -25 -0.25 0.15 -0.375 -5 -0.05 0.15 -0.0075 0 0 0.3 0 25 0.25 0.2 0.05 65 0.65 0.2 0.13 Expected return 0.135 Both firms have an expected return of 0.135, meaning that they will make 13.5% back from their investments. b.) Suppose that a single investor has only one unit to invest. Which type of firm would the investor prefer to invest in and why? (6 points) It doesn ’ t matter which firm the investor chooses since the probability of making money for both firms are the same at a 30% chance of losing money and a 70% change of making a profit. c.) Suppose the venture fund has 100 units to invest. If it decides to invest them ALL in Type A firms, find the expected average return on ALL 100 investments and compute its standard deviation. (6 points) Type A Firms % Returns X P(x) X*P(x) (X-mean) ^2*P(x) -15 -0.15 0.3 -0.045 0.02437 0 0 0.3 0 0.005468 45 0.45 0.4 0.18 0.03969 Mean, variance 0.135 0.06953 The mean is 0.135. The standard deviation is square root(0.06953) = 0.2637. d.) Between b and c above which would be more likely to yield the expected return? Why? (4 points)
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