Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribute of the number of spades. Hence, find the mean of the distribution.

Explanation

We have,

`p=` probability of getting a spade in a draw` =frac{13}{52}=frac{1}{4}` and

`q=1-p=1-frac{1}{4}=frac{3}{4}`

Let` X` denote a success of getting a spade in a throw .
Then,
`{X}` follows binomial distribution with parameters `{n}=3 and p=frac{1}{4}`
` therefore P(X=r)=^{3} C_{r} p^{r} q^{(3-r)}=^{3} C_{r}(frac{1}{4})^{r}(frac{3}{4})^{(3-r)}`
`=frac{^{3} C_{r} 3^{(3-r)}}{4^{3}}`
`=frac{27}{64}(frac{^{3} C_{r}}{3^{r}}), where , r=0,1,2,3`

So, the probability distribution of `X` is given by:

`P(X=r)=frac{27}{64}(frac{{ }^{3} C_{r}}{3^{r}}), where, r=0,1,2,3`

Now mean,`E(X)=np=3 times 1/4=3/4`