a.
Find the
a.

Answer to Problem 36E
The expected value and standard deviation of total winnings when a gambler bets $3 on a single round are –0.081 and 2.9989, respectively.
Explanation of Solution
From the given information at a university, there are 37 slots with 18 red slots, 18 black slots, and 1 green slot. The gambler can place their bets on red or black; if the ball lands on their color, they double their money and if the ball lands on other color, they lose their money.
There are a total of 37 outcomes in that there are18 chances that the ball lands on a red, and there are 18 chances that the ball lands on the black color. There is one chance that the ball lands on the green color.
The expected value is calculated as follows:
Thus, the expected value is –0.081.
Standard deviation:
Thus, standard deviation is 2.9989.
b.
Find the expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds.
b.

Answer to Problem 36E
The expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds are –0.081 and 173, respectively.
Explanation of Solution
From the given information at a university, there are 37 slots with 18 red slots, 18 black slots, and 1 green slot. The gambler can place their bets on red or black; if the ball lands on their color, they double their money and if the ball lands on other color, they lose their money.
There are a total of 37 outcomes in that there are18 chances that the ball lands on the red and there are 18 chances that ball lands on the black color. There is one chance that the ball lands on the green color.
The expected value is calculated as follows:
The expected value of the total winning for a single round is $–0.027.
The expected value for three rounds is calculated as follows:
Thus, the expected value is $–0.081.
Standard deviation:
For three rounds
Thus, standard deviation is 1.73.
c.
Compare the results obtained from Part (a) and Part (b) and describe the riskiness of two games.
c.

Answer to Problem 36E
The expected values for both the games are the same but Part (b) game has a less standard deviation than Part (a).
The bet of $3 on a single round is riskier than bet of $1 in three different rounds.
Explanation of Solution
The results obtained in Part(a) and Part(b) are given as follows:
The expected value and standard deviation of total winnings when a gambler bets $3 on a single round are –0.081 and 2.9989, respectively.
The expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds are –0.081 and 1.73, respectively.
The expected values for both the games are the same, but Part (b) games have a less standard deviation than Part (a).
Thus, a bet of $3 on a single round is riskier than a bet of $1 in three different rounds.
Want to see more full solutions like this?
Chapter 3 Solutions
OPENINTRO:STATISTICS
- Theorem 2.6 (The Minkowski inequality) Let p≥1. Suppose that X and Y are random variables, such that E|X|P <∞ and E|Y P <00. Then X+YpX+Yparrow_forwardTheorem 1.2 (1) Suppose that P(|X|≤b) = 1 for some b > 0, that EX = 0, and set Var X = 0². Then, for 0 0, P(X > x) ≤e-x+1²² P(|X|>x) ≤2e-1x+1²² (ii) Let X1, X2...., Xn be independent random variables with mean 0, suppose that P(X ≤b) = 1 for all k, and set oσ = Var X. Then, for x > 0. and 0x) ≤2 exp Σ k=1 (iii) If, in addition, X1, X2, X, are identically distributed, then P(S|x) ≤2 expl-tx+nt²o).arrow_forwardTheorem 5.1 (Jensen's inequality) state without proof the Jensen's Ineg. Let X be a random variable, g a convex function, and suppose that X and g(X) are integrable. Then g(EX) < Eg(X).arrow_forward
- Can social media mistakes hurt your chances of finding a job? According to a survey of 1,000 hiring managers across many different industries, 76% claim that they use social media sites to research prospective candidates for any job. Calculate the probabilities of the following events. (Round your answers to three decimal places.) answer parts a-c. a) Out of 30 job listings, at least 19 will conduct social media screening. b) Out of 30 job listings, fewer than 17 will conduct social media screening. c) Out of 30 job listings, exactly between 19 and 22 (including 19 and 22) will conduct social media screening. show all steps for probabilities please. answer parts a-c.arrow_forwardQuestion: we know that for rt. (x+ys s ا. 13. rs. and my so using this, show that it vye and EIXI, EIYO This : E (IX + Y) ≤2" (EIX (" + Ely!")arrow_forwardTheorem 2.4 (The Hölder inequality) Let p+q=1. If E|X|P < ∞ and E|Y| < ∞, then . |EXY ≤ E|XY|||X|| ||||qarrow_forward
- Theorem 7.6 (Etemadi's inequality) Let X1, X2, X, be independent random variables. Then, for all x > 0, P(max |S|>3x) ≤3 max P(S| > x). Isk≤narrow_forwardTheorem 7.2 Suppose that E X = 0 for all k, that Var X = 0} x) ≤ 2P(S>x 1≤k≤n S√2), -S√2). P(max Sk>x) ≤ 2P(|S|>x- 1arrow_forwardThree players (one divider and two choosers) are going to divide a cake fairly using the lone divider method. The divider cuts the cake into three slices (s1, s2, and s3).If the chooser's declarations are Chooser 1: {s3} and Chooser 2: {s3}, which of the following is a fair division of the cake?arrow_forwardTheorem 1.4 (Chebyshev's inequality) (i) Suppose that Var X x)≤- x > 0. 2 (ii) If X1, X2,..., X, are independent with mean 0 and finite variances, then Στη Var Xe P(|Sn| > x)≤ x > 0. (iii) If, in addition, X1, X2, Xn are identically distributed, then nVar Xi P(|Sn> x) ≤ x > 0. x²arrow_forwardTheorem 2.5 (The Lyapounov inequality) For 0arrow_forwardTheorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
- Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningHolt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL