An urn contains 5 red and 2 black balls. Two balls are randomly selected. Let X represent the number of black balls. What are the possible values of X? Is X a random variable?

Question

Philoid · Accepted Answer

The key point to solve the problem:

A variable X is said to be a random variable if the sum of probabilities associated with each value of X gets equal to 1

i.e. ∑(p_i) = 1 where p_i is probability associated with x_i

X represents the number of black balls drawn.

∴ X can take values 0,1 or 2 as both balls drawn can be black which corresponds to X = 2, either of 2 balls can be black which corresponds to X = 1, and if neither of balls drawn is black it corresponds to X = 0

∵ there are a total of 7 balls

n(S) = total possible ways of selecting 2 balls =

P(X = 0) = P(selecting no black balls) =

P(X = 1) = P(selecting 1 black ball and 1 red)

=

P(X = 0) = P(selecting all black balls) =

Now we have p_i and x_i.

Clearly, ∑(p_i) =

∴ X is a random variable, and the above table represents its probability distribution.