Question

There are 32 games played in the first round of the NCAA basketball tournament. Joe enters a pool where you choose the winner of each game. He knows nothing about basketball and picks his teams randomly by placing both team’s names on slips of paper. Puts them in a hat and draws one out. The team he pulls out is the team he chooses to win the game. Let X be the number of picks Joe gets correct out of the 32 first-round games.

A. Find ?(? = 15).

B. Find ?(? ≤ 12)

C. Find ?(13 ≤ ? ≤ 18).

D. Would you be surprised if Joe got more than 22 out of 32 correct using his method? Explain

Answer 1

For each of the 32 games, Joe puts the names of the 2 teams in a hat and randomly picks one as the winner. The probability that he picks the winner of any given game is 0.5.

Let X be the number of picks Joe gets correct out of the 32 games. We can say that X has a Binomial distribution with parameters, number of trials (number of games) n=32 and success probability (The probability that he picks the winner of any given game) p=0.5

The probability that Joe gets X=x correct is

A. Find ?(? = 15).

ans: ?(? = 15)=0.1317

B. Find ?(? ≤ 12)

ans: ?(? ≤ 12) = 0.1077

C. Find ?(13 ≤ ? ≤ 18).

ans: ?(13 ≤ ? ≤ 18) = 0.7038

D. Would you be surprised if Joe got more than 22 out of 32 correct using his method? Explain

The probability that Joe got more than 22 out of 32 correct using his method is 0.01. This means, the probability of Joe getting more than 22 out of 32 correct using his method is less than 0.05

Hence Joe getting more than 22 out of 32 correct using his method is a rare event.

ans: Yes, we would  be surprised if Joe got more than 22 out of 32 correct using his method.