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)=\dfrac{C_{11}^2}{C_{16}^2}=\dfrac{\dfrac{11!}{2!(11-2)!}}{\dfrac{16!}{2!(16-2)!}}=\dfrac{\dfrac{10\cdot 11}{2}}{\dfrac{15\cdot 16}{2}}=\dfrac{55}{120}=\dfrac{11}{24}.](https://tex.z-dn.net/?f=Pr%28A%29%3D%5Cdfrac%7BC_%7B11%7D%5E2%7D%7BC_%7B16%7D%5E2%7D%3D%5Cdfrac%7B%5Cdfrac%7B11%21%7D%7B2%21%2811-2%29%21%7D%7D%7B%5Cdfrac%7B16%21%7D%7B2%21%2816-2%29%21%7D%7D%3D%5Cdfrac%7B%5Cdfrac%7B10%5Ccdot%2011%7D%7B2%7D%7D%7B%5Cdfrac%7B15%5Ccdot%2016%7D%7B2%7D%7D%3D%5Cdfrac%7B55%7D%7B120%7D%3D%5Cdfrac%7B11%7D%7B24%7D.)
B. The probability that neither selected team has won at least 1 super bowl is
![Pr(B)=\dfrac{C_{5}^2}{C_{16}^2}=\dfrac{\dfrac{5!}{2!(5-2)!}}{\dfrac{16!}{2!(16-2)!}}=\dfrac{\dfrac{4\cdot 5}{2}}{\dfrac{15\cdot 16}{2}}=\dfrac{10}{120}=\dfrac{1}{12}.](https://tex.z-dn.net/?f=Pr%28B%29%3D%5Cdfrac%7BC_%7B5%7D%5E2%7D%7BC_%7B16%7D%5E2%7D%3D%5Cdfrac%7B%5Cdfrac%7B5%21%7D%7B2%21%285-2%29%21%7D%7D%7B%5Cdfrac%7B16%21%7D%7B2%21%2816-2%29%21%7D%7D%3D%5Cdfrac%7B%5Cdfrac%7B4%5Ccdot%205%7D%7B2%7D%7D%7B%5Cdfrac%7B15%5Ccdot%2016%7D%7B2%7D%7D%3D%5Cdfrac%7B10%7D%7B120%7D%3D%5Cdfrac%7B1%7D%7B12%7D.)
C. The probability that at least one selected team has won at least 1 super bowl is
![Pr(C)=1-Pr(B)=1-\dfrac{1}{12}=\dfrac{11}{12}.](https://tex.z-dn.net/?f=Pr%28C%29%3D1-Pr%28B%29%3D1-%5Cdfrac%7B1%7D%7B12%7D%3D%5Cdfrac%7B11%7D%7B12%7D.)
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)=\dfrac{Pr(P\cap Q)}{Pr(Q)}=\dfrac{\dfrac{5\cdot 11}{C_{16}^2}}{\dfrac{C_5^1\cdot C_{15}^1}{C_{16}^2}}=\dfrac{55}{75}=\dfrac{11}{15}.](https://tex.z-dn.net/?f=Pr%28P%7CQ%29%3D%5Cdfrac%7BPr%28P%5Ccap%20Q%29%7D%7BPr%28Q%29%7D%3D%5Cdfrac%7B%5Cdfrac%7B5%5Ccdot%2011%7D%7BC_%7B16%7D%5E2%7D%7D%7B%5Cdfrac%7BC_5%5E1%5Ccdot%20C_%7B15%7D%5E1%7D%7BC_%7B16%7D%5E2%7D%7D%3D%5Cdfrac%7B55%7D%7B75%7D%3D%5Cdfrac%7B11%7D%7B15%7D.)
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)=\dfrac{Pr(M\cap N)}{Pr(N)}=\dfrac{\dfrac{11\cdot 10}{C_{16}^2}}{\dfrac{C_{11}^1\cdot C_{15}^1}{C_{16}^2}}=\dfrac{110}{165}=\dfrac{2}{3}.](https://tex.z-dn.net/?f=Pr%28M%7CN%29%3D%5Cdfrac%7BPr%28M%5Ccap%20N%29%7D%7BPr%28N%29%7D%3D%5Cdfrac%7B%5Cdfrac%7B11%5Ccdot%2010%7D%7BC_%7B16%7D%5E2%7D%7D%7B%5Cdfrac%7BC_%7B11%7D%5E1%5Ccdot%20C_%7B15%7D%5E1%7D%7BC_%7B16%7D%5E2%7D%7D%3D%5Cdfrac%7B110%7D%7B165%7D%3D%5Cdfrac%7B2%7D%7B3%7D.)