Question

a jar contains 11 red marbles, 12 blue marbles, and 6 white marbles. marbles from this jar are selected, with each marble being replaced after each selection. what is the probability that the first white marble chosen is on the 4th selection? 0.3957 0.6043 0.5011 0.1032

Explanation:

Step1: Find total number of marbles

Total marbles = 11 (red) + 12 (blue) + 6 (white) = 29.

Step2: Probability of not white in first 3 selections

Probability of not white (red or blue) = (11 + 12)/29 = 23/29. Since selections are with replacement, each trial is independent. So probability of not white in first 3: $( \frac{23}{29} )^3$.

Step3: Probability of white on 4th selection

Probability of white = 6/29. Multiply with step2 result: $P = ( \frac{23}{29} )^3 \times \frac{6}{29}$.
Calculate:
$(23^3) = 23\times23\times23 = 12167$, $29^3 = 24389$, so $(23/29)^3 = 12167/24389$.
Then multiply by 6/29: $(12167\times6)/(24389\times29) = 73002/707281 \approx 0.1032$? Wait, no, wait my calculation was wrong. Wait 23+12=25? Wait no, 11 red +12 blue is 23? Wait 11+12=23, yes. 23 non - white, 6 white. Total 29.
Wait let's recalculate:
$(23/29)^3\times(6/29)=\frac{23^3\times6}{29^4}$
23^3 = 12167, 29^4 = 29×29×29×29 = 29×29=841, 841×29 = 24389, 24389×29 = 707281
12167×6 = 73002
73002÷707281≈0.1032? But wait the options have 0.1032. Wait but maybe I made a mistake in non - white count. Wait 11 red +12 blue is 23, yes. 6 white. Total 29.
Wait another way: Probability of not white is (11 + 12)/29 = 23/29. So three non - white: (23/29)^3, then white: 6/29. So (23/29)^3*(6/29). Let's compute 23÷29≈0.7931, 0.7931^3≈0.7931×0.7931 = 0.6290, 0.6290×0.7931≈0.499, then 0.499×(6/29)≈0.499×0.2069≈0.1032. Yes, so the probability is approximately 0.1032.

Wait but the first option I thought was wrong, but according to calculation, it's 0.1032. Wait but let's check again. Wait 11 red, 12 blue, 6 white. 11+12=23, 23+6=29. Correct. So the probability that first white is on 4th selection is the probability of 3 non - white and then white. So formula for geometric distribution: $P(X = k)=(1 - p)^{k - 1}p$, where p is probability of success (white), k = 4. Here p = 6/29, 1 - p = 23/29. So $P(X = 4)=(23/29)^{3}\times(6/29)\approx0.1032$.

Wait but the options have 0.1032 as one of them. So the correct answer is 0.1032.

Answer:

0.1032