There are two simultaneous games. One game is flipping a coin. Payment of tail is $2.50 and head is $1.50. And
the other game is rolling a six sided die and paying number of dollars shown on the face of the die. Both the coin
and the die are determined simultaneously and are independent of each other. Payment is the product of the two
resulting games. What is the expected amount of payment in this joint game?
To find the expected amount of payment in this joint game, we need to calculate the expected value for each game separately and then multiply them together.
First, let's calculate the expected value for the coin-flipping game:
- The probability of getting tails is 1/2, and the payment for tails is $2.50.
- The probability of getting heads is also 1/2, and the payment for heads is $1.50.
So, the expected value for the coin-flipping game can be calculated as follows:
Expected value = (Probability of tails * Payment for tails) + (Probability of heads * Payment for heads)
Expected value = (1/2 * $2.50) + (1/2 * $1.50)
Expected value = $1.25 + $0.75
Expected value = $2.00
Next, let's calculate the expected value for the die-rolling game:
- Since it is a fair six-sided die, each number (1, 2, 3, 4, 5, 6) has an equal probability of 1/6.
- The payment for each number shown on the die is equal to that number.
So, the expected value for the die-rolling game can be calculated as follows:
Expected value = (Probability of rolling a 1 * Payment for rolling a 1) + (Probability of rolling a 2 * Payment for rolling a 2) + ... + (Probability of rolling a 6 * Payment for rolling a 6)
Expected value = (1/6 * 1) + (1/6 * 2) + (1/6 * 3) + (1/6 * 4) + (1/6 * 5) + (1/6 * 6)
Expected value = (1/6)(1 + 2 + 3 + 4 + 5 + 6)
Expected value = (1/6)(21)
Expected value = 3.5
Finally, to calculate the expected amount of payment in the joint game, we multiply the expected values of the two games together:
Expected amount of payment = Expected value of coin-flipping game * Expected value of die-rolling game
Expected amount of payment = $2.00 * 3.5
Expected amount of payment = $7.00
Therefore, the expected amount of payment in this joint game is $7.00.