Question

Urn problem One jar contains 8 white marbles and 12 black marbles. We offer you the following three games. For each game, identify the probability law(distribution) that will follow the number of points you will earn, the expected value and its Standard deviation.

A) You draw ten marbles and earn one point for each white ball drawn. (with replacement)

B)You draw ten marbles and earn one point for each white ball drawn. (Without replacement)

C) You draw marbles (with replacement) until you draw a black one and earn ten points for each white marble drawn.

Answer 1

A) The number of trials is finite , n= 10

If marbles are drawn with replacement , probability of drawing white marble in each draw remains same, p =8/12

Let X be the number of points earned

X follow Binomial with n = 10 , p = 0.67

Probability mass function of X is

, x =0,1,2...,8

Mean = np = 10*0.67 = 6.7

Standard deviation =

B)The sampling is done from a finite population(N=12+8 =20) , and drawing of a white marble from K= 8 white marbles is considered success

As the sampling is done without replacement , the probability of drawing white marble in each draw keeps changing

This is the case of hypergeometric distribution

Let X be the number of points won

X follow Hypergeometric distribution with N= 20 , n =10 , K =8

Then probability mass function of X is

, x = 0,1,2,....8

mean = nK/N = 10*8/20 = 4

standard deviation =

C) Let X be the number of points won until a black marble is drawn

That is X is the number of white marbles *10 , before the first black marble

Here drawing a black marble is considered success , probability of success is constant , p = 12/20 = 0.6

and drawing white marbles are considered failure

X follow Geometric distribution distribution with p= 0.6

Probability mass function of X is

, x = 0,10, 20 ,..., 80

mean = (1-p) /p = 0.4/ 0.6 = 0.67

standard deviation=