Six dice are thrown `729` time. How many times do you expect at least three dice to show a five or six.

Explanation

Let,

`P=` probability of showing five or six `=frac{2}{6}=frac{1}{3}`

`n=6, r=3`

`q=1-p=1-frac{1}{3}=frac{2}{3}`

`P(X=r) =1-[P(X=0)+P(X=1)+P(X=2)]`

`=1-[C_0^6 (frac{2}{3})^6 (frac{1}{3})^0+C_1^6 (frac{2}{3})^5 (frac{1}{3})^1+C_2^6 (frac{2}{3})^2 (frac{1}{3})^4]`

`=1-[ (frac{2}{3})^6 (frac{1}{3})^0+6 (frac{2}{3})^5 (frac{1}{3})^1+15 (frac{2}{3})^2 (frac{1}{3})^4]`

`=1-(frac{2}{3})^4[ frac{4+12+15}{9}]`

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