Question

Consider a game of chance consisting of a single trial with exactly two outcomes, which from a players perspective we will call "win" and "lose." To play the game, a player must wager an amount, which we will denote by a. If the player loses the game, a player loses their wager. If the player wins the game, then they keep their wager and they win $1.00. Denote the probability by p, where 0 < p < 1. Let the random variable X denote the amount won by the player.

A) Find the sample space of the random variable X.

B) Find the pmf of the distribution of the random variable X.

C) Compute the expression for E(X), the expected value of X.

D) A game is said to be fair if the expected amount won is 0. For what value of a, the amount wagered, would the game be described as a fair game?

E) For what vaules of a is E(X) >0?

F) For what values of a is E(X) <0?

G) Suppose a person is only willing to play if their expected amount won is non negative. For what values of a would this person be willing to play, and what values of a would this person not be willing to play?

Answer 1

a: Wager Amount

p : Probability of win

X: the amount won by the player;

A) The sample space of the random variable X.

The possible values of

X = 1 (Win : wager amount + $1)

X = -a (losing wager amount)

Therefore sample space of X : (a+1, -a)

(B) the pmf of the distribution of the random variable X

p : Probability to win

1-p : Probability to lose

Player wins : X = 1 ; probability (X=1) = p

Player loses : X = -a ; probability (X=-a) = 1-p

PMF of X :

P(X) = p f or X= 1

P(X) = 1-p for X=-a

C)

Expected value of X

D)For a fair game E(X) = 0

p-a+ap = 0

p=a-ap

a(1-p) = p

a = p / (1-p)

For a fair game , a = p/(1-p)

E) for what values of 'a' E(X) > 0

p-a+ap > 0

p +ap-a > 0

ap-a > -p

a(p-1) > -p

a > -p/(p-1)

a > -p/-(1-p)

a > p/(1-p)

if a > p/(1-p) then E(X) > 0

F)

for what values of 'a' E(X) < 0

p-a+ap < 0

p +ap-a < 0

ap-a < -p

a(p-1) < -p

a < -p/(p-1)

a < -p/-(1-p)

a < p/(1-p)

if a < p/(1-p) then E(X) < 0

G)Suppose a person is only willing to play if their expected amount won is non negative.

i.e E(X)

i.e

if a p/(1-p) --- this person be willing to play

if E(X) < 0 then the person not be willing to play

i.e if a < p/(1-p) then the person not be willing to play.