Let X be the number of running plays. X has the binomial distribution with n = 8 trials and success probability of 15/31.

In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, …, n

P[X = x] = 0 for any other value of x.

P(of at least 3 passing plays)

= P( at most 5 running plays)

= P(X ≤ 5 ) = P(X = 0) + P(X = 1) + P(X = 2) + … + P(X =5)

P(X = 0) = 0.005035775

P(X = 1) = 0.037768312

P(X = 2) = 0.123927273

P(X = 3) = 0.232363638

P(X = 4) = 0.272301138

P(X = 5) = 0.204225853

P(X = 6) = 0.095730869

P(X = 7) = 0.025642197

P(X = 8) = 0.003004945

P(X ≤ 5 ) = 0.875622

P(at least two running plays)

P(X ≤ 2 ) = P(X = 0) + P(X=1) + P(X=2)

= 0.1667314