. In a roulette game, the ball is equally likely to fall in any one of the 38 slots. For three bets of $5 each on Black on separate spins of the roulette wheel, find

(a) the probability distribution of the daughter gamble S₃by completing the following table;

Table of Computations for Distribution of S₃

A	B	C	D	E	F
Outcome (s3)	Underlying Sequences (x1, x2, x3)	Sequence Probability (conceptually)	Sequence Probability (value)	Outcome Probability (conceptually)	Outcome Probability (value)
15	5, 5 ,5	(18/38)3 (20/38)0	?	1 * ?
5	5, 5, -5	(18/38)2 (20/38)1	??	3 * ??

(b) From your table, compute the expected value of S₃;

(c) From your table, compute the variance of S₃.

(d) Are the results in (b) and (c) in accordance with the following equations?

Important Facts About the Sum of Gambles

Let X and Y denote two gambles (or investments). Then, the expected value of the daughter gamble given by the sum of the bets, denoted X+Y, is equal to the sum of the expected values of the parent gambles:

E(X+Y) = E(X) + E(Y).

If the two parent bets are independent, then the variance of the daughter gamble is equal to the sum of the variances of the parent gambles:

V(X+Y) = V(X) + V(Y).

Now, consider a player who makes n independent gambles on the same parent game. Those bets are called independent and identical, or i.i.d. An example for n=2 would be 2 bets of $5 each on Black. Let S_n denote the sum of these n i.i.d. bets. And, let X denote any one of those parent bets. Then it can readily be shown from the above rules that

E(S_n) = n E(X).

V(S_n) = n V(X),

_Sn = n _X.

These three rules should look hauntingly similar to those found on page 9, but there are important differences. The present rules have to do with the sums of gambles, while the rules on page 9 were concerned with changing the size of the wager for a single gamble.