Question

The Laplace High School basketball team has a probability of 1/2 of

winning a game, independent of previous games. The probability

that, in a sequence of 8 games, the team wins two of its last four

games and four of its eight games can be expressed in lowest

terms as 7. What is a + b?

Answer 1

a= 1/2 and b= 1/2

Explanation:

The given probability can be expressed in the form of binomial expansion where a= p and b= q

Therefore a= 1/2 and b= 1/2

For n= 8 trials the binomial expansion can be written as

[ p+q] ^n = [ 1/2+1/2]^8

=(1/2)^0(1/2)^8 +8C1(1/2)^7(1/2)+8C2(1/2)^6(1/2)^2+8C3(1/2)^5(1/2)^3+8C4(1/2)^4(1/2)^4+8C5(1/2)^3(1/2)^5+8C6(1/2)^2(1/2)^6+8C7(1/2)(1/2)^7+8C8(1/2)^0(1/2)^8

=

1(1/2)^0(1/2)^8 +8(1/2)^7(1/2)+28(1/2)^6(1/2)^2+56(1/2)^5(1/2)^3+70(1/2)^4(1/2)^4+56(1/2)^3(1/2)^5+28(1/2)^2(1/2)^6+8(1/2)(1/2)^7+1(1/2)^0(1/2)^8

=

1/(2)⁸ { 1+ 8+28+56+70+56+28+8+1}

= 1/(2)⁸ {256}

= 1

The total probability is always 1.

Also P+q= 1

1/2+1/2= 1