Question

Miguel is playing a game in which a box contains four chips with numbers written on them. Two of the chips have the number 1, one chip has the number 3, and the other chip has the number 5. Miguel must choose two chips, and if both chips have the same number, he wins $2. If the two chips he chooses have different numbers, he loses $1 (–$1). • Let X = the amount of money Miguel will receive or owe. Fill out the missing values in the table. (Hint: The total possible outcomes are six because there are four chips and you are choosing two of them.) Xi 2 –1 P(xi) • What is Miguel’s expected value from playing the game? • Based on the expected value in the previous step, how much money should Miguel expect to win or lose each time he plays? • What value should be assigned to choosing two chips with the number 1 to make the game fair? Explain your answer using a complete sentence and/or an equation.

Answer 1

See explanation

Explanation:

Step 1

Miguel wins $2 is if he pulls two chips with the number 1.

Probability of winning is:

• 2/4 * 1/3 = 1/6 as there are 4 chips in total

Probability of loosing is:

• 1- 1/6 = 5/6

Step 2

Missing values we found in the step 1, can be populated in the table:

• Xi === 2 === -1
• P(Xi) === 1/6 === 5/6

Expected Value as per table data:

• 1/6*2 + 5/6*(-1) = -1/2

Expected value is $1/2 loss each time he plays

Step 3

To make the game fair, the expected value should be zero.

Then as per the calculation above, let's replace 2, with x, and find its value.

• 1/6x + 5/6*(-1) = 0
• 1/6x = 5/6
• x= 5

So the amount should be $5 to make the game fair.