An urn contains 4 white and 3 red balls. Three balls are drawn at random from the urn with replacement. Find the probability distribution of the number of red balls.

Explanation

As three balls are drawn with replacement, the number of white balls, say `X`, follows binomial distribution with `n=3`

`p=frac{3}{7} and q=frac{4}{7}`

`P(X=r)=^{3} C_{r}(frac{3}{7})^{r}(frac{4}{7})^{3-r}, r=0,1,2,3`

`P(X=0)=^{3} C_{0}(frac{3}{7})^{0}(frac{4}{7})^{3-0}`

`P(X=1)=^{3} C_{1}(frac{3}{7})^{1}(frac{4}{7})^{3-1}`

`P(X=2)=^{3} C_{2}(frac{3}{7})^{2}(frac{4}{7})^{3-2}`

`P(X=3)=^{3} C_{3}(frac{3}{7})^{3}(frac{4}{7})^{3-3}`

For`X = \0, \ 1,\ 2,\ 3 `
` P(X)\ frac{64}{343} \ frac{144}{343}\ frac{108}{343} \ frac{27}{343}`