Question

You have a bag of marbles. Your bag contains 75 marbles. You only want the red marbles. You know that 30 of the marbles in the bag are red. You randomly select two marbles from the bag and set them aside without looking at them. Let X represent the number of red marbles removed. (hint. This is without replacement.)

a) find the probability mass function (i.e. probability distribution) for the number of red marbles selected.

b) find the expected number of red marbles selected.

c) what is the probability fewer than two red marbles are removed from the bag?

Answer 1

Total marbles in bag= 75

Red marbles in bag = 30

random variable X = no. of red marbles selected out of 2 marbles selected

Possible values of X =0 , 1 , 2

Formula for conditional probability gives :  

here P(B|A) is the conditional probability of Event B occuring given that Event A has already occured .

this implies :

a) Probability mass function of X is the Probability of various values of X .

(i) P(X=0)=Probability that no red marble is selected

=P[(no red marble on first draw) and (no red marble in second draw)]

= P(no red marble on first draw ) * P(no red marble on second draw given no red marble on first draw)

P(no red marble on first marble ) =

Since for second draw , total marbles left are 74 and no . of red marbles left are 30.

P(no red marble on second draw given no red marble on first draw) =   

(ii) P(X=1) =P[( red marble on first draw) and (no red marble in second draw)] +P[(no red marble on first draw ) and (red marble on second draw)]

  P(red marble on first and no red marble on second ) =

P(no red marble in first and red marble on second) =

(iii) P(X=2) =P[( red marble on first draw) and ( red marble on second draw)]

= P( red marble on first draw ) * P( red marble on second draw given red marble on first draw)

  =

=

P(X) = Probability mass fuction is as follows :

x	0	1	2
P(X=x)

b) Expected value of a discrete random variable is given as :

  

So

c) Probability that fewer than two red marbles are removed = P(X<2)

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