Question

A jar contains 11 red marbles, 12 blue marbles, and 6 white marbles. four marbles from this jar are selected, with each marble being replaced after each selection. what is the standard deviation of x, the number of draws until the first red marble?
0.4852
0.9704
2.0770
1.5172

Answer 1

The PMF of $X$ is almost geometric in nature. Let $p=\frac{11}{29}$. Then

$P(X = x) = \begin{cases} p & \text{if }x = 0 \\ (1-p)p & \text{if }x = 1 \\ (1-p)^2 p & \text{if }x = 2 \\ (1-p)^3p & \text{if }x = 3 \\ (1-p)^4 & \text{if }x = 4 \\ 0 & \text{otherwise}\end{cases}$

Compute the first moment/expected value.

$E(X) = \displaystyle \sum_x x\, P(X=x) \\\\ ~~~~~~~~ = 0\cdot p+1\cdot(1-p)p + 2\cdot(1-p)^2p + 3\cdot(1-p)^3p + 4\cdot(1-p)^4 \approx 1.39349$

Compute the second moment.

$E(X^2) = \displaystyle \sum_x x^2\, P(X=x) \\\\ ~~~~~~~~ = 0\cdot p+1\cdot(1-p)p + 4\cdot(1-p)^2p + 9\cdot(1-p)^3p + 16\cdot(1-p)^4 \approx 4.01103$

Compute the variance.

$V(X) = E(X^2) - E(X)^2 \approx 2.06921$

The standard deviation is the square root of the variance.

$\sqrt{V(X)} \approx \boxed{1.43848}$