Question

Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable `X` denotes the number of hearts in the three cards drawn. determine the probability distribution of X.

Answer 1

Correct Answer - `"Mean "=(3)/(4), "variance "=(9)/(16)`
There are 13 hearts and 39 other cards.
Let E= event of drawing a heart.
Then, `P(E )=(13)/(52)=(1)/(4) and P(barE)=(1-(1)/(4))=(3)/(4).`
Let X= number of hearts in a draw.
Then, X=0,1,2 or 3.
`P(X=0)=P(barE barE barE)=P(barE)xxP(barE)xxP(barE)=((3)/(4)xx(3)/(4)xx(3)/(4))=(27)/(64).`
`P(X=1)=P[(EbarEbarE) or (barEbarE E)]`
`=P(EbarEbarE)+P(barE EbarE)+P(barEbarE E)`
`=((1)/(4)xx(3)/(4)xx(3)/(4))+((3)/(4)xx(1)/(4)xx(3)/(4))+((3)/(4)xx(3)/(4)xx(1)/(4))=(27)/(64).`
`P(X=2)=P[(E E barE) or (E barE E) or (barE E E)]`
`=P(E E barE)+P(E barE E)+P(barE E E)`
`=((1)/(4)xx(1)/(4)xx(3)/(4))+((1)/(4)xx(3)/(4)xx(1)/(4))+((3)/(4)xx(1)/(4)xx(1)/(4))=(9)/(64).`
`P(X=3)=P( E E E)=P(E)xxP(E)xxP(E)=((1)/(4)xx(1)/(4)xx(1)/(4))=(1)/(64).`

Find the mean and variance.