Question

There are currently 16 teams in the National Football Conference (NFC). Of those teams, 11 have won at least one super bowl and 5 have not. Suppose that 2 teams are selected at random without replacement. a. Find the probability that both selected teams have won at least 1 super bowl. b. Find the probability that neither selected team has won at least 1 super bowl. c. Find the probability that at least one selected team has won at least 1 super bowl. d. Find the probability that the second team selected has won at least 1 super bowl given that the first team selected has not won a super bowl. e. Find the probability that the second team selected has won at least 1 super bowl given that the first team selected has won at least 1 super bowl.

Answer 1

You are given 16 teams, 11 have won at least one super bowl and 5 have not.

A. The probability that both selected teams have won at least 1 super bowl is

`Pr(A)=(C_(11)^2)/(C_(16)^2)=((11!)/(2!(11-2)!))/((16!)/(2!(16-2)!))=((10\cdot 11)/(2))/((15\cdot 16)/(2))=(55)/(120)=(11)/(24).`

B. The probability that neither selected team has won at least 1 super bowl is

`Pr(B)=(C_(5)^2)/(C_(16)^2)=((5!)/(2!(5-2)!))/((16!)/(2!(16-2)!))=((4\cdot 5)/(2))/((15\cdot 16)/(2))=(10)/(120)=(1)/(12).`

C. The probability that at least one selected team has won at least 1 super bowl is

`Pr(C)=1-Pr(B)=1-(1)/(12)=(11)/(12).`

D. to find the probability that the second team selected has won at least 1 super bowl given that the first team selected has not won a super bowl, consider such events:

P - the second team selected has won at least 1 super bowl;

Q - the first team selected has not won a super bowl.

Then

`Pr(P|Q)=(Pr(P\cap Q))/(Pr(Q))=((5\cdot 11)/(C_(16)^2))/((C_5^1\cdot C_(15)^1)/(C_(16)^2))=(55)/(75)=(11)/(15).`

E. To find the probability that the second team selected has won at least 1 super bowl given that the first team selected has won at least 1 super bowl, consider events:

M - the second team selected has won at least 1 super bowl;

N - the first team selected has won at least 1 super bowl.

Then

`Pr(M|N)=(Pr(M\cap N))/(Pr(N))=((11\cdot 10)/(C_(16)^2))/((C_(11)^1\cdot C_(15)^1)/(C_(16)^2))=(110)/(165)=(2)/(3).`