200 CHAPTER 4 Discrete Probability Distributions Finding an Expected Value In Exercises 37 and 38, find the expected value E1x2 to the player for one play of the game. If x is the gain to a player in a game of chance, then E1x2 is usually negative. This value gives the average amount per game the player can expect to lose. 37. In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, . . ., 34, 35, and 36, marked on equally spaced slots. If a player bets $1 on a number and wins, then the player keeps the dollar and receives an additional $35. Otherwise, the dollar is lost. 38. A high school basketball team is selling $10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas valued at $5460, and the second prize is a weekend ski package valued at $496. The remaining 18 prizes are $100 gas cards. The number of tickets sold is 3500. Extending Concepts 39. Writing Find the area of each bar of the histogram you made in Exercise 19. Then find the sum of the areas. Interpret the results. 40. Baseball There were 116 World Series from 1903 to 2020. Use the probability distribution in Exercise 30 to find the number of World Series that had 4, 5, 6, 7, and 8 games. Find the population mean, variance, and standard deviation of the data using the traditional definitions. Compare to your answers in Exercise 30. Linear Transformation of a Random Variable In Exercises 41 and 42, use this information about linear transformations. For a random variable x, a new random variable y can be created by applying a linear transformation y = a + bx, where a and b are constants. If the random variable x has mean mx and standard deviation sx, then the mean, variance, and standard deviation of y are given by the formulas my = a + bmx, sy 2 = b2 sx 2, and sy = 0 b0 sx. 41. The mean annual salary of employees at an office is originally $46,000. Each employee receives an annual bonus of $600 and a 3% raise (based on salary). What is the new mean annual salary (including the bonus and raise)? 42. The mean annual salary of a firm’s employees is $44,000 with a variance of 18,000,000. What is the standard deviation of the salaries after each employee receives an annual bonus of $1000 and a 3.5% raise (based on salary)? Independent and Dependent Random Variables Two random variables x and y are independent when the value of x does not affect the value of y. When the variables are not independent, they are dependent. A new random variable can be formed by finding the sum or difference of random variables. If a random variable x has mean mx and a random variable y has mean my, then the means of the sum and difference of the variables are given by mx+y = mx + my and mx-y = mx - my. If random variables are independent, then the variance and standard deviation of the sum or difference of the random variables can be found. So, if a random variable x has variance s 2 x and a random variable y has variance s 2 y, then the variances of the sum and difference of the variables are given by s 2 x+y = s 2 x + s 2 y and s 2 x-y = s 2 x + s 2 y. In Exercises 43 and 44, the distribution of SAT mathematics scores for college-bound male seniors in 2020 has a mean of 531 and a standard deviation of 121. The distribution of SAT mathematics scores for college-bound female seniors in 2020 has a mean of 516 and a standard deviation of 112. One male and one female are randomly selected. Assume their scores are independent. (Adapted from College Board) 43. Find the mean and standard deviation of the sum of their scores. 44. Find the mean and standard deviation of the difference of their scores. Compare to your answers in Exercise 43.
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