Answer:
a) The expected value is ![\frac{-1}{15}](https://tex.z-dn.net/?f=%5Cfrac%7B-1%7D%7B15%7D)
b) The variance is ![\frac{49}{45}](https://tex.z-dn.net/?f=%5Cfrac%7B49%7D%7B45%7D)
Step-by-step explanation:
We can assume that both marbles are withdrawn at the same time. We will define the probability as follows
#events of interest/total number of events.
We have 10 marbles in total. The number of different ways in which we can withdrawn 2 marbles out of 10 is
.
Consider the case in which we choose two of the same color. That is, out of 5, we pick 2. The different ways of choosing 2 out of 5 is
. Since we have 2 colors, we can either choose 2 of them blue or 2 of the red, so the total number of ways of choosing is just the double.
Consider the case in which we choose one of each color. Then, out of 5 we pick 1. So, the total number of ways in which we pick 1 of each color is
. So, we define the following probabilities.
Probability of winning: ![\frac{2\binom{5}{2}}{\binom{10}{2}}= \frac{4}{9}](https://tex.z-dn.net/?f=%5Cfrac%7B2%5Cbinom%7B5%7D%7B2%7D%7D%7B%5Cbinom%7B10%7D%7B2%7D%7D%3D%20%5Cfrac%7B4%7D%7B9%7D)
Probability of losing ![\frac{(\binom{5}{1})^2}{\binom{10}{2}}\frac{5}{9}](https://tex.z-dn.net/?f=%5Cfrac%7B%28%5Cbinom%7B5%7D%7B1%7D%29%5E2%7D%7B%5Cbinom%7B10%7D%7B2%7D%7D%5Cfrac%7B5%7D%7B9%7D)
Let X be the expected value of the amount you can win. Then,
E(X) = 1.10*probability of winning - 1 probability of losing =![1.10\cdot \frac{4}{9}-\frac{5}{9}=\frac{-1}{15}](https://tex.z-dn.net/?f=1.10%5Ccdot%20%20%5Cfrac%7B4%7D%7B9%7D-%5Cfrac%7B5%7D%7B9%7D%3D%5Cfrac%7B-1%7D%7B15%7D)
Consider the expected value of the square of the amount you can win, Then
E(X^2) = (1.10^2)*probability of winning + probability of losing =![1.10^2\cdot \frac{4}{9}+\frac{5}{9}=\frac{82}{75}](https://tex.z-dn.net/?f=1.10%5E2%5Ccdot%20%20%5Cfrac%7B4%7D%7B9%7D%2B%5Cfrac%7B5%7D%7B9%7D%3D%5Cfrac%7B82%7D%7B75%7D)
We will use the following formula
![Var(X) = E(X^2)-E(X)^2](https://tex.z-dn.net/?f=%20Var%28X%29%20%3D%20E%28X%5E2%29-E%28X%29%5E2)
Thus
Var(X) = ![\frac{82}{75}-(\frac{-1}{15})^2 = \frac{49}{45}](https://tex.z-dn.net/?f=%5Cfrac%7B82%7D%7B75%7D-%28%5Cfrac%7B-1%7D%7B15%7D%29%5E2%20%3D%20%5Cfrac%7B49%7D%7B45%7D)