Question

In Exercises 5–36, express all probabilities as fractions.

Win \(1 Billion Quicken Loans offered a prize of \)1 billion to anyone who could correctly predict the winner of the NCAA basketball tournament. After the “play-in” games, there are 64 teams in the tournament.

a. How many games are required to get 1 championship team from the field of 64 teams?

b. If you make random guesses for each game of the tournament, find the probability of picking the winner in every game.

Short answer

a. The number of games required to get 1 championship team when a total of 64 teams are playing is equal to 63.

b. The probability of picking the winner in every game is equal to$\frac{1}{2^{63}}$.

Step-by-step solution

Given information

A basketball tournament is played between 64 teams.

Methods for counting

The number of possibilities in which an event can happen are counted according to the given situation.

Either the entire set of possible combinations can also be listed and counted or rules for permutation and combination would be used.

Compute the number of games

Number of teams is 64.

Matches will be played with two teams at a time and the successively winning teams would decide for the ultimate winner.

Step 1:

The number of matches played between 64 teams with 2 teams in each match is equal to 32.Each of the 32 matches will have 32 winners.

Step 2:

The number of matches played between the 32 winning teams with 2 teams in each match = 16

Each of the 16 matches will have 16 winners.

Step 3:

The number of matches played between the 16 winning teams with 2 teams in each match = 8

Each of the 8 matches will have 8 winners.

Step 4:

The number of matches played between the 8 winning teams with 2 teams in each match = 4

Each of the 4 matches will have 4 winners.

Step 5:

The number of matches played between the 4 winning teams with 2 teams in each match = 2

Each of the 2 matches will have 2 winners.

Step 6:

One final match will be played between the 2 winning teams to decide the winner of the tournament.

The total number of matches played is equal to:

$32+16+8+4+2+1=63$

Thus, the total number of games required for obtaining 1 championship team is equal to 63.

Compute the number of games

b.

Out of 2 teams in one match, 1 will win.

The probability of choosing the winning team in any one match is equal to $\frac{1}{2}$.

The total number of matches played is 63.

The probability of choosing the winning team in all the 63 games is given as follows:

$$\begin{gathered} \frac{1}{2}\times \frac{1}{2}\times .....\times \frac{1}{2}\left(\text{upto}\text{63}\text{times}\right)={\left(\frac{1}{2}\right)}^{63} \\ =\frac{1}{2^{63}} \end{gathered}$$

Thus, the probability of choosing the winning team for all the 63 games is equal to $\frac{1}{2^{63}}$.