Question

You have a bag with 5 marbles. 4 are black and 1 is red. You draw marbles without replacement and stop once you get the red marble. a) What is the expected number of draws? b) What is the variance in the number of draws?

Answer 1

Let X be the number of draws until the red marble is got. Then the probability distribution is given by:

X	P(X=x)
1	=1/5	0.2
2	=4/5*1/4	0.2
3	=4/5*3/4*1/3	0.2
4	=4/5*3/4*2/3*1/2	0.2
5	=4/5*3/4*2/3*1/2*1	0.2

a)

The expected number of draws is:

b)

Table:

X	P(X=x)	x*P(X=x)	(x-E(X))^2	((x-E(X))^2)*P(X=x)
1	0.2	0.2	4	0.8
2	0.2	0.4	1	0.2
3	0.2	0.6	0	0
4	0.2	0.8	1	0.2
5	0.2	1	4	0.8
	Sum	3		2

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