5. A bag contains ten marbles, 5 are yellow, 3 are blue and 2 are red. An experi

Question

5. A bag contains ten marbles, 5 are yellow, 3 are blue and 2 are red. An experiment is carried out that consists of three trials. In each trial, a marble is drawn from the bag at random. When a yellow marble is drawn from the bag it is called a success; the removal of any other marble is a failure. After each trial, the marble is replaced.

(a) Is the experiment binomial? Justify your answer.

(b) What is the probability that no yellow marble is drawn in the three trials?

(c) What is the expected number of yellow marbles drawn in the three trials?

(d) Suppose that we made one change to the experiment, which was that we did not replace the marbles after each trial. Would this new experiment be binomial? Justify your answer

Explanation / Answer

a. The experiment is binomial as there are only 2 options, either success or failure, and the probability of each is a constant as we consider trials with replacement

b. As marbles are replaced after each trial, probability of success and failure is the same in all trials.

Prob. of success = 5/10 = 0.5

Prob. of failure = 1-0.5 = 0.5

Prob. of no yellow marbles in 3 trials = 3C0*0.5^3 = 0.125

c. Expected number of yellow marbles in 3 trials is:

sum x.p(x)= 0+0.375+0.750+0.375 = 1.5 = E(X)

So, expected number of yellow marbles in 3 trials = 1.5

d. Now the probability of success and failure is different in each trail and hence the process is not binomial

Marbles(x) Probabilityp(x) x.p(x) 0 0.125 0 1 3C1*0.5^3 0.375 2 3C2*0.5^3 0.75 3 0.125 0.375

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