Question

2. In a roulette game, the ball is equally likely to fall in any one of the 38 slots. You will bet on 4 numbers. A roulette bet on 4 numbers has a payoff of 8:1. You have doubled your bet from $5 to $10.

     (a) Show the probability distribution of the $10 bet as a table;

(b) Graphically, show how the probability distribution of the doubled bet is related to the original bet of $5;

(c) Compute the expected value of the doubled bet from (a);

(d) Check that the expected value of the doubled bet is in accordance with the expected value of a discrete random variable theory, involving the expected value of the $5 bet;

(e) Compute the variance of the $10 bet from the probability distribution in (a);

(f) Check that the results of (e) are properly related to the variance of the $5 bet.

Answer 1

prob of winning==>1/38

prob of loosing==>37/38

winning price==> $80

lossing price==>$10

Answer a:

probability distribution table:

X	80	-10
P(X=x)	1/38	37/38

Answer b:

Answer C:

E( x ) ==> 80*( 1/38 ) + (- 10*37/38)

==> - 7.6

d) E(10$ bet) ==> (4/38)*10*8 - (34/38)*10 ==> 10*(-2/38)

E(5$ bet) ==> 5*(-2/38). So E(10$ bet) ==> 2*E(5$ bet). This is in accordance with expected value of discrete random variable theory. [Property of linearity)

e) Var(10$ bet) ==> 4/38 * 80* 80 - (34/38) *100 - (-20/38)^2 ==> 583.9335

f) Var(5$ bet) ==> 4/38 * 40*40 - (34/38)*25 - (10/38)^2 ==> 145.983

Var(10$ bet) is roughly 4*Var(5$ bet). Hence properly related to variance of 5$ bet

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