Question

There are 49 marbles in a jar. 7 are red, 7 are green, 7 are purple, 7 are yellow, 7 are orange, 7 are black, and 7 are white. (a) You randomly pull out 6 marbles from the jar with replacement (so you draw one, put it back, draw another, put it back, and so on). Compute the probability that you pull out atleast 1 red marble. (b) [Let X be a random variable representing the number of different colors of marbles that you end up drawing (i.e. if you pull out 3 orange marbles, 2 green marbles, and 1 black marble, then you have pulled out 3 distinct colors and X = 3). We can express X as the sum of 7 Bernoulli random variables X = Xr + Xg + Xp + Xy + Xo + Xb + Xw, where Xr = (1 if at least 1 red marble is pulled 0 if no red marbles are pulled , and Xg, Xp, Xy, Xo, Xb, and Xw are defined similarly for the other 6 colors. Are the 7 Bernoulli random variables Xr, Xg, Xp, Xy, Xo, Xb and Xw independent? Explain intuitively why or why not and then prove your answer mathematically. (c) Compute E(X). (d) Find all values N such that if you draw N marbles with replacement, you will draw at least 4 marbles of the same color with probability 1. Justify your answer.

Answer 1

Answer:

You randomly pull out 6 marbles from the jar with replacement (so you draw one, put it back, draw another, put it back, and so on). Compute the probability that you pull out atleast 1 red marble. (b) [Let X be a random variable representing the number of different colors of marbles that you end up drawing (i.e. if you pull out 3 orange