Question

4 singles, 7 fives, 3 twenties, and 2 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?

Answer 1

For this to be a fair game, the price to play should match the expected value of the prize.

To calculate the expected value we just need to use the formula for the expected value, which is

`E=\sum ^n_(i=1)P_{}(x_i)x_i`

Where x_i represents the values of our dataset, and P(x_i) the probability of this value to be choosen from the hat.

To find the probability of drawing each individual bill, we just need to calculate the ratio between the amount of this specific bill and the total amount of bills inside of the hat.

We have 4 $1 bills, 7 $5 bills, 3 $20 bills, and 2 $100 bills. The total amount of bills is given by the sum of all of those amounts.

`4+7+3+2=16`

Then, the probabilities for each value are

`\begin{gathered} P(1)=(4)/(16)=(1)/(4) \\ P(5)=(7)/(16) \\ P(20)=(3)/(16) \\ P(100)=(2)/(16)=(1)/(8) \end{gathered}`

Then, applying our values into the expected value formula, we have

`\begin{gathered} E=(1)/(4)\cdot1+(7)/(16)\cdot5+(3)/(16)\cdot20+(1)/(8)\cdot100 \\ E=(1)/(4)+(35)/(16)+(60)/(16)+(100)/(8) \\ E=0.25+2.1875+3.75+12.5 \\ E=18.6875\approx18.69 \end{gathered}`

The fair price for this game would be $18.69.