Suppose that for high school football games played in the ra

Suppose that, for high school football games played in the rain, 60% are won by the home team, 20% feature fewer than four touchdowns, and 37% feature at least one injury.

a.If a random sample of 10 high school football games that were played in the rain is selected, what is the probability that between 2 and 4 (including “exactly 2” and “exactly 4”) of them will have featured fewer than four touchdowns?

b.If a random sample of 11 high school football games that were played in the rain is selected, what is the probability that exactly 3 of the selected games will have featured at least one injury.

Solution

a)
P( X = 2 ) = ( 10 2 ) * ( 0.2^2) * ( 1 - 0.2 )^8
= 0.302
P( X = 3 ) = ( 10 3 ) * ( 0.2^3) * ( 1 - 0.2 )^7
= 0.2013
P( X = 4 ) = ( 10 4 ) * ( 0.2^4) * ( 1 - 0.2 )^6
= 0.0881
P(2 < = X < = 4) = 0.302+0.2013+0.0881 = 0.5914
b)
P( X = 3 ) = ( 11 3 ) * ( 0.37^3) * ( 1 - 0.37 )^8
= 0.2074