A die is rolled 12 times Find the probability of rolling no

A die is rolled 12 times. Find the probability of rolling no more than 3 fours.

(type a decimal rounded to four decimals as needed. round all intermediate values to four decimal places as needed.)

Solution

Let X be the number of fours obtained when a die is rolled.
X follows a binomial distribution where the number of trials n is 12 and he probability of a success p is 1/6.

So X~ B( 12, 1/6)

The probability mass function of X is,

P(X=x) = (n C x) p^x (1-p)^n-x

Now we have to find the probability of rolling no more than 3 fours.

that is P(X <=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

P(X=0) = (12C0) (1/6)⁰ (1-1/6)¹² = 0.1122

P(X=1) = (12C1) (1/6)¹ (1-1/6)¹¹ = 0.2692

P(X=2) = (12C2) (1/6)² (1-1/6)¹⁰ = 0.2961

P(X=3) = (12C3) (1/6)³ (1-1/6)⁹ = 0.1974

P(X <=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

= 0.1122 + 0.2692 + 0.2961 + 0.1974 = 0.8749