Find the probability of getting 5 exactly twice in 7 throws of a die.

Explanation

Let X represent the number of times of getting 5 in 7 throws of the die.

Probability of getting 5 in a single throw of a die, p=`1/6`​

`q=1−p=1−1/6​=5/6​`

Clearly, X has the probability distribution with n=7 and p=1/6
​
P(X=x)=`n_(c_x )q^(n−x)p^x=7_(c_x )​(5/6)^(7−x)⋅(1/6​)^x`

P(getting 5 exactly twice)=P(X=2)

=`7_(c_2 )​(5/6​)^5⋅(1/6​)^2`

=`21⋅(5/6​)^5⋅1/36`

...