You draw marbles from a jar containing 5 black and 95 grey marbles. Let X be...

You draw marbles from a jar containing 5 black and 95 grey marbles. Let X be the number of drawings (with replacement) it takes until you get your first black marble. Let Y be the number of drawings (with replacement) it takes until you get your fifth black marble. Let Z be the number of black marbles removed if you draw 10 marbles without replacement.

Calculate E[X] , Calculate P (Y = 90) ,Calculate Var(Z).

Solutions

Expert Solution

a) The expected value of the random variable X here is computed as:

= 1/p

Where p is the probability of getting a black card on any of the drawing

Therefore 20 is the expected value here.

b) The probability here is computed as:

P(Y = 90) = Probability that there are 4 black cards drawn in the first 89 draws* Probability that the fifth black card is drawn on the 90th draw

Therefore 0.009751 is the required probability here.

c) As we are drawing cards without replacement here, therefore this is a case of hypergeometric distribution with the following parameters:

N = 100 as the population size,
K = 5 as the number of successes in the population,
n = 10 is the sample size.

Therefore now the variance here is computed as:

Therefore 0.4318 is the required variance here

orchestra answered 1 year ago

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