Chapter 9, Problem 16PS

(a)

To determine

To find: The expected value of the $\$1$ ticket.

Answer to Problem 16PS

There expected value is $-0.71$ .

Explanation of Solution

Given:

The expected value V of an event A is the sum of the products of the outcomes’ probabilities ( $p_1,p_2,\dots ,p_n$ ) and their values ( $x_1,x_2,\dots ,x_n$ ),

  $V=p_1{x}_1+p_2{x}_2+\dots +p_n{x}_n$

Calculation:

The probability of selecting 5 different numbers chosen from the numbers 1 to 47 is $C{}_5$ .

The probability of selecting one MEGA number chosen from the numbers 1 to 27 is 27.

So, the combined probability of choosing 5 different numbers from a lot of 1 to 47 and choosing one MEGA number from a lot of 1 to 27 numbers is $C{}_5\times 27$ .

The jackpot for the next drawing is $\$12,000,000$ .

So, the expected value of the $\$1$ ticket is calculated as follows:

  $\begin{array}{c}V=p_1{x}_1+p_2{x}_2\\ =\left(\frac{1}{C{}_5\times 27}\times 12,000,000\right)+\left(1-\frac{1}{C{}_5\times 27}\times \left(-1\right)\right)\\ \approx -0.71\end{array}$