Question

5 singles, 8 fives, 4 twenties, and 3 hundred dollar bills are all placed in a hat. If a player is to reach into the hat and randomly choose one bill, what is the fair price to play this game? State your answer in terms of dollars rounded to the nearest cent (hundredth).

Answer 1

The fair price to play the game is $23.61

Step-by-step explanation:

Number of singles = 5

number of fives = 8

Number of twenties = 4

number of $300 = 1

Total number = 5 + 8 + 4 + 1 = 18

fraction for each:

5/18 chance of getting singles

8/18 chance of getting fives

4/18 chance of getting twenties

1/18 chance of getting $300

We find the Expected values:

`\begin{gathered} \text{singles = 1} \\ \text{Expected }value\text{ = (}(5)/(18)\text{ }*1)\text{ +(}\frac{\text{ 8}}{18}*\text{ 5)+ (}\frac{\text{4}}{18}*20)\text{ + (}\frac{\text{1}}{18}*300) \\ \text{Expected }value\text{ =}(1)/(18)\text{ \lbrack(}5\text{ }*1)\text{ +(}8*\text{ 5)+ (}4*20)\text{ + (}1*300)\rbrack \\ \text{Expected value =}(1)/(18)\text{(}5\text{ + 40 + 80 + 300)}_{} \\ \\ \text{fair price = }\frac{Total\text{ money in the hat}}{total\text{ number of bills}} \\ \text{fair price = }(1)/(18)\text{(}5\text{ + 40 + 80 + 300)}_{} \end{gathered}`
`\begin{gathered} \text{fair price = }(1)/(18)\text{(}425\text{)}_{} \\ \text{fair price = \$23.61} \end{gathered}`

In the absence of further information, fair price is $23.61