Answer:
a) E(X) = -$0.0813 , s.d (X) = 3
b) E(X) = -$0.0813 , s.d (X) = 3
c) expected loss and higher stakes of loosing.
Step-by-step explanation:
Given:
- There are total 37 slots:
Red = 18
Black = 18
Green = 1
- Player on bets on either Red or black
- Wins double the bet money, loss the best is lost
Find:
a) Expected value of earnings X if we place a bet of $3
b) Expected value and standard deviation if we bet $1 each on three rounds
c) compare the two answers in a and b and comment on the riskiness of the two games
Solution:
- Define variable X as the total winnings per round. We will construct a distribution tables for total winnings per round for bets of $3 and $ 1:
- Bet: $3
X -3 3 E(X)
P(X) 1-0.48 = 0.5135 18/37 = 0.4864 3*(.4864-.52135) = -0.08
-The s.d(X) = sqrt(9*(0.5135 + 0.4864) - (-0.08)^2) = 3.0
- Bet: $1
X -1 1 E (X)
P(X) 1-0.48 = 0.5135 18/37 = 0.4864 1*(.4864-.5135) = -0.0271
-The s.d(X) = sqrt(1*(0.5135 + 0.4864) - (-0.0271)^2) = 0.999
- The expected value for 3 rounds is:
E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3)
- The above X winnings are independent from each round, hence:
E(3*X_1) = 3*E(X_1) = 3*(-0.0271) = -0.0813
- The standard deviation for 3 rounds is:
sqrt(Var(X_1 + X_2 + X_3)) = sqrt(Var(X_1) + Var(X_2) + Var(X_3))
- The above X winnings are independent from each round, hence:
sqrt(Var(3*X_1)) = 3*Var(X_1) = 3*(0.999) = 2.9988
- For above two games are similar with an expected loss of $0.0813 for playing the game and stakes are very high due to high amount of deviation for +/- $3 of winnings.