Chapter 10.4, Problem 7E

Solution

To calculate: The money per day for the carnival and also find carnival is making money or losing money.

The money per day for the carnival is $75 and carnival is making money.

Given information:

The Great Flubini is correct 72% of the time, off from 5 to 10 lb 21% of the time, and off by more than 10 lb 7% of the time. If The Great Flubini guesses the weights of 300 visitors per day, on average

Formula used:

For calculating expected value or mean using the following formula:

  $\mu ={\displaystyle \sum xP\left(x\right)}$   ...... (1)

Calculation:

When the Great Flubini guesses their weight within 5 lb, then no prize is given while $2 was paid by the person and thus a profit of $2 was then made by the carnival. Moreover, the guess is within 5 lb 72% of the time.

  $P\left(-\$3\right)=21\%=0.21$

When the Great Flubini is off from 5 to 10 lb, then the participant gets a prize of $5 while the participant paid $2 and thus a loss of $5-$2=$3 was then made by the carnival. Moreover, the guess is off from 5 to 10 lb about 21% of the time.

  $P\left(-\$3\right)=21\%=0.21$

When the Great Flubini is off by more than 10 lb, then the participant gets a prize of $10 while the participant paid $2and thus a loss of $10-$2=$8 was then made by the carnival. Moreover, the guess is off by more than 10 lb about 7% of the time.

  $P\left(-\$8\right)=7\%=0.07$

The expected value (or mean) is the sum of the product of each possibility x with its probability P(x):

  $\mu ={\displaystyle \sum xP\left(x\right)}$

Substitute $x\text{and}P\left(x\right)$ for different cases in equation (1) respectively,

  $\begin{array}{c}\mu =\$2\times 0.72+\left(-\$3\right)\times 0.21+\left(-\$8\right)\times 0.07\\ =\$1.44-\$0.63-\$0.56\\ =\$0.25\end{array}$

Thus, the carnival is making money as the expected value of $0.25 is positive Moreover, they are expected to make $0.25 per person.

In this case, we have 300 people and thus the carnival is expected to make 300×$0.25 = $75 per day.