Question

In a box of 5 balls, 2 are red and 3 are blue. Two balls are randomly selected (without replacement). Let X be the number of red balls in the two selected balls.

a. Find the probability distribution of X (i.e., list all possible values of X and their corresponding probabilities).

b. Find the expected value and the standard deviation of X.

Answer 1

At the start, we have 5 balls out of which 2 are red and 3 are blue

Therefore the probability of red ball = 2/5 = 0.4

probability of blue ball = 3/5 = 0.6

Now we are randomly selecting 2 balls and X is the random variable denoting the number of red balls in those 2 balls.

Therefore X follows the binomial distribution with parameters n and p.

Here n =2

p = probability of drawing red ball = 0.4

a) Now X has three possible values 0,1 and 2.

P(X=0)

= 0.36

P(X=1)

= 0.48

P(X=2)

=0.16

Therefore probability distribution of X is

X	P(X)
0	0.36
1	0.48
2	0.16

b) In binomial distribution

Expectation = = 2 * 0.4 = 0.8

Variance = = 2* 0.4*0.6 = 0.48

Standard deviation =   = = 0.6928