An urn contains 5 red 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable ? if yes, find the mean and variance of X.

Answer

Mean, `E(X)=4/7` and Variance, `var(X)=50/147`

Explanation

Let `X` denote the number of black balls drawn


Total Number of balls= red balls+black balls`=5+2=7`


`therefore X " can take values " 0,1,2`


Now the probability of drawing red cards are:


` P(X=0)=P("no black ball drawn")`


`=(5C_2 times 2C_0)/(7C_2)=(frac{5!}{2!3!} times frac{2!}{0!2!})/(frac{7!}{2!5!})`


`=(frac{5 times 4 times 3!}{2 times 3!} times frac{2!}{2!})/(frac{7 times 6 times 5!}{2 times 5!})=frac{5 times 2}{7 times 3}=10/21`


`rArr P(X=0)=10/21`


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