answer:
To find the amount a casino should charge a gambler to play one round of this game so that the expected value to the casino is $1.25, we can follow these steps:
1. Calculate the probability of each possible outcome:
There are two bills selected at random from the bag. Let's consider the possible outcomes:
- Two $100 bills: The probability of selecting the first $100 bill is 10/70, and the probability of selecting the second $100 bill is 9/69. So the probability of this outcome is (10/70) * (9/69).
- Two $1 bills: The probability of selecting the first $1 bill is 60/70, and the probability of selecting the second $1 bill is 59/69. So the probability of this outcome is (60/70) * (59/69).
- One $100 bill and one $1 bill: There are two possible cases here, either the $100 bill is selected first or the $1 bill is selected first. The probability of selecting a $100 bill first and then a $1 bill is (10/70) * (60/69). The probability of selecting a $1 bill first and then a $100 bill is (60/70) * (10/69). So the total probability of this outcome is [(10/70) * (60/69)] + [(60/70) * (10/69)].
2. Calculate the expected value for the casino:
The expected value is the sum of each outcome multiplied by its probability. Let's denote the amount charged by the casino as x.
The expected value for the casino can be calculated as follows:
(2 * $100 - x) * [(10/70) * (9/69)] + (2 * $1 - x) * [(60/70) * (59/69)] + (($100 + $1) - x) * {[(10/70) * (60/69)] + [(60/70) * (10/69)]} = $1.25
3. Solve the equation for x:
Simplify the equation and solve for x to find the amount the casino should charge the gambler.
4. Verify the solution:
Once you find the value of x, substitute it back into the equation and ensure that the expected value to the casino is indeed $1.25.
In summary, to find the amount a casino should charge a gambler to play one round of this game so that the expected value to the casino is $1.25, we need to calculate the probabilities of each outcome, set up the equation for the expected value, solve for the amount charged (x), and verify the solution.
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