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

Explanation:

Step1: Calculate the probability of success (getting a red marble)

The total number of marbles is \(11 + 12+6=29\). The probability of getting a red marble \(p=\frac{11}{29}\approx0.3793\).

Step2: Use the formula for the standard deviation of a geometric distribution

The formula for the standard deviation of a geometric distribution \(X\) (number of trials until the first success) is \(\sigma=\sqrt{\frac{1 - p}{p^{2}}}\).
Substitute \(p = \frac{11}{29}\) into the formula:
First, \(1-p=1-\frac{11}{29}=\frac{29 - 11}{29}=\frac{18}{29}\).
Then \(\sigma=\sqrt{\frac{\frac{18}{29}}{(\frac{11}{29})^{2}}}=\sqrt{\frac{18\times29}{11^{2}}}=\sqrt{\frac{522}{121}}\approx\sqrt{4.314}\approx2.0770\)

Answer:

2.0770