In: Statistics and Probability
A roulette wheel has numbers from 1 to 36 and two "special" numbers 0 and 00. A $1 bet on a number will pay $36 if that number comes up. It is also possible to bet on two adjacent numbers, with a $18 payout if one of them spells, or four adjacent numbers, with a $9 payout.
a) Let X be the total payout of a $1 bet on a number. Find E(X)
and Var(X).
b) Let Y be the total payoff of a $1 bet on two adjacent numbers.
Find E(Y ) and Var(Y ).
c) Let Z be the total payoff of a $1 bet on four adjacent numbers.
Find E(Z) and Var(Z).
d) As a general rule, a $1 bet that will win in ''n'' out of 38
will pay 36/n $ . Find the mathematical expectation and variance of
the total payoff of a $1 bet that would win in ''n'' cases.
Part a
part b and c
Here I have assumed that either if one ,two or three or all of the adjacent chosen number come then also it could be considered a win.