# Expectation and Fair price

#### aburchett

##### New member
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 ##### Elite Member 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 ##### New member I'm so sorry, the spinner looks like this: [attachment=0:15s6zp88]spinner.png[/attachment:15s6zp88] #### soroban ##### Elite Member Hello again, aburchett! Okay, much better . . . A person spins the pointer and is awarded the amount indicated by the pointer. Code:  * * * * | * * | * * |$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.