Answer :
Answer:
[tex] A=5 *\frac{1}{4} +10 *\frac{1}{4} + 0 *\frac{1}{2}=\frac{5}{4} +\frac{10}{4}=\frac{15}{4}=3.75[/tex]
Step-by-step 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:
[tex] E(X) = \sum_{i=1}^n X_i P(X_i)[/tex]
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.
[tex] P(diamond) = 13/52=\frac{1}{4}[/tex]
[tex]P(heart) = 13/52=\frac{1}{4}[/tex]
[tex]P(black)= 26/52=\frac{1}{2}[/tex]
And we have the following values:
[tex] X_{diamond}= 5, X_{heart}=10, X_{black}=0[/tex]
Let A the amount that we need to pay for a game we have this equality:
[tex] 5 *\frac{1}{4} +10 *\frac{1}{4} + 0 *\frac{1}{2}-A=0[/tex]
And if we solve for A we got:
[tex] A=5 *\frac{1}{4} +10 *\frac{1}{4} + 0 *\frac{1}{2}=\frac{5}{4} +\frac{10}{4}=\frac{15}{4}=3.75[/tex]