Question

A game is played by drawing a single card from a regular deck of playing cards. If you get a black card, you win

nothing. If you get a diamond, you win $5.00. If you get a heart, you win $10.00. How much would you be willing
to pay if the game is to be fair? Explain.

Answer 1

`A=5 *(1)/(4) +10 *(1)/(4) + 0 *(1)/(2)=(5)/(4) +(10)/(4)=(15)/(4)=3.75`

Explanation:

Previous concepts

The expected value of a random variable X is the n-th moment about zero of a probability density function f(x) if X is continuous, or the weighted average for a discrete probability distribution, if X is discrete.

Solution to the problem

For this case we need to use the definition of expected value given by:

`E(X) = \sum_(i=1)^n X_i P(X_i)`

For this case we need that the expected value would be 0 since we want a fair game.

Assuming a standard deck of 52 cards. We can find the probabilities for each possible event.

`P(diamond) = 13/52=(1)/(4)`

`P(heart) = 13/52=(1)/(4)`

`P(black)= 26/52=(1)/(2)`

And we have the following values:

`X_(diamond)= 5, X_(heart)=10, X_(black)=0`

Let A the amount that we need to pay for a game we have this equality:

`5 *(1)/(4) +10 *(1)/(4) + 0 *(1)/(2)-A=0`

And if we solve for A we got:

`A=5 *(1)/(4) +10 *(1)/(4) + 0 *(1)/(2)=(5)/(4) +(10)/(4)=(15)/(4)=3.75`