Expectation and Fair price

[aburchett]

A person spins the pointer and is awarded the amount indicated by the pointer.



It costs $8 to play the game. Determine:


· The expectation of a person who plays the game.


· The fair price to play the game.

My work:
E=P(wins)(amount won)+P(loss)(amount lost)
E=1/3(-6)+2/3(-8)
E=1/3*-6/1 + 2/3*-8/1
E=-2/1+-16/3
E=-6/3+-16/3
E=-22/3=7.333
E=$7.33

Fair price=expectation + cost of play
Fair price= 7.33 +8.00
Fair price= $15.33

This just doesn't seem right to me, can someone tell me where I went wrong?

 

[soroban]

Hello, aburchett!

You left out a lot of details . . . like:
. . how is the spinner divided?
. . what are the payoff?

A person spins the pointer and is awarded the amount indicated by the pointer.

It costs $8 to play the game.

Determine:

. . (a) The expectation of a person who plays the game.
. . (b) The fair price to play the game.

My work:

\(\displaystyle E\:=\(\text{win})\cdot (\text{amount won}) \,+\,P(\text{lose})\cdot(\text{amount lost})\)

\(\displaystyle E \:=\:\left(\tfrac{1}{3}\right)(\text{-}6) \,+\,\left(\tfrac{2}{3}\right)(\text{-}8)\) . ??

Do I understand the situation?

Two-thirds of the time I lose $6.
The other one-third of the time, I lose $8.

\(\displaystyle \text{My expectation is: }\:-2-\frac{16}{3} \:=\:-\frac{22}{3}\)

\(\displaystyle \text{I can expect to }lose\text{ an average of }\$7.33\text{ per game.}\)


And you think I'll pay $8 to play this game . . . I don't think so!

 

[aburchett]

I'm so sorry, the spinner looks like this:
[attachment=0:15s6zp88]spinner.png[/attachment:15s6zp88]

 

[soroban]

Hello again, aburchett!

Okay, much better . . .

A person spins the pointer and is awarded the amount indicated by the pointer.

            * * *
        *     |     *
      *       |       *
     *        |   $8   *
              |
    *         |         *
    *   $2    * - - - - *
    *         |         *
              |
     *        |  $12   *
      *       |       *
        *     |     *
            * * *

It costs $8 to play the game.

Determine:

(a) The expectation of a person who plays the game.

(b) The fair price to play the game.

We can expect the following:

. . \(\displaystyle \begin{array}{c}\text{win \$2 with probability }\tfrac{1}{2} \\ \\[-3mm] \text{win \$8 with probability }\tfrac{1}{4} \\ \\[-3mm] \text{win \$12 with probability }\tfrac{1}{4} \end{array}\)


If we played for free, our expected value would be:

. . \(\displaystyle E \;=\;\left(\tfrac{1}{2}\right)(2) + \left(\tfrac{1}{4}\right)(8) + \left(\tfrac{1}{4}\right)(12) \;=\; 6\)

We could expect to win an average of $6 per game.

(a) Since we pay $8 to play each game, we have an average loss of $2 per game.


(b) The fair price to pay is $6 per game.