You have a jar that contains black marbles and gold marbles. There are a total o

Question

You have a jar that contains black marbles and gold marbles. There are a total of 10 marbles in the jar. A trial consists of taking out a marble at random and then replacing it.

You conduct a total of m trials. The expected number of black marbles you observe during the m trials is 2.8 and the variance of the number of black marbles you observe during the m trials is 0.84.

a) Determine m and the composition of marbles in this jar

You then put the marbles back into the jar. Your friend comes along. She conducts the same trials, and records the number of trials she observes the first gold marble.

b) What is the probability that she observes the first gold marble on the fifth trial given that the first gold marble appears on an odd-numbered trial?

c) What is the variance of the number of trials on which you observe the first gold marble?

Explanation / Answer

Solution-

(a) Here let us denote the number of black marbels drawn by the random variable X. It must be a binomial random variable and given that

mean = m*p = 2.8 and variance = m* p *(1-p) = 0.84

Solving above two equations, we get -

m = 4 and p = 0.7

(b) Let the number of trials on which you observe the first gold marble be a random variable Y, which must follow negative binomial distribution with parameter k = 1 and p = 1-0.7 = 0.3

Required probability = P(Y=5)

= 4c0 * 0.30 * 0.74 * 0.3

= 0.07203

(c) var(Y) = k*(1-p)/p2

= 1*0.7 / 0.32

= 7.7778

Answer

TY!

=

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