Question

13. A bin contains 3 red and 4 green balls. 3 balls are chosen at random, with replacement. Let the random variable X be the number of green balls chosen. a. Explain why X is a binomial random variable. b. Construct a probability distribution table for X. c. Find the mean (expected value) of X. d. Use the law of Large Numbers to interpret the meaning of the expected value of X in the context of this problem.

Answer 1

13. a) There are 7 balls ( 3 red and 4 green balls )

X be the number of green balls out of 3 balls chosen

n= 3 balls are chosen , that is finite number of trials

As the balls are drawn with replacement , each draw is independent of other

thus probability of success ( that is probability of drawing a green ball) is constant in each trail , p=4/7

Thus X is Binomial random variate .

b) Probability mass function of X is

, x=0,1,2,3

For x = 0 , probability is

similarly we calculate for x= 1,2,3

Probability distribution table for X is

x	P(x)
0	0.0787
1	0.3149
2	0.4198
3	0.1866

c) E(X) = np = 3*4/7 = 1.7

d) According to law of large numbers , the average of the results obtained from a large number of experiments should be close to the expected value and gets even closer as the number of experiments are increased .

  Thus if we repeat the experiment of drawing 3 balls large number of times , the average number of green balls drawn will be closer to 1.7.