The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions.
2. One evening, Fyodor decides to play 10 games of roulette, betting $100 on black each time (this bet wins $100 with probability 18/38, and otherwise loses $100).
(i) What is the expected value and standard deviation of his total winnings?
(ii) Find the exact chance (i.e., not using a normal approximation) that he loses $400 or more over the course of the evening.
(i) This is a diachonomial distribution.
N = 10
P = 18/38 = 0.474
Then E(Winning) = N*P = 4.74
And the expected earning = 100*4.74 - 100*(10-4.74) = -$52
For a single trial, the expected value is M = 100*0.474 - 100*(1-0.474) = -5.2
variance is V = P*(100-M)^2 + (1-P)*(-100-M)^2
V = 0.474*(100+5.2)^2 + ...
This solution is comprised of a detailed explanation to answer what is the expected value and standard deviation of his total winnings and find the exact chance (i.e., not using a normal approximation) that he loses $400 or more over the course of the evening.