Consider a simple game in which you will roll a six-sided die and receive $3.00 if you roll a 6. If it costs you $0.90 for each roll, what is the expected value?

You gain $3 if you get a 6, and always pay $0.90.

E[X] = (1/6)*3 - $0.90

prob of rolling a 6 = 5/36

( ways to get a 6 : [5,1 ; 4,2 ; 3,3 ; 2,4 ; 1,5 ] )

expected value = $3.00(5/36) = .41666 or 42 cents

Would you pay 90 cents for a game that has an expected value of 42 cents?

To calculate the expected value, we need to multiply the probability of each outcome by its corresponding value and sum them up.

In this case, there are six possible outcomes when rolling a six-sided die: 1, 2, 3, 4, 5, and 6. The probability of rolling each number is 1/6 since the die is fair.

The value of rolling a 6 is $3.00, while the cost of each roll is $0.90.

Now, let's calculate the expected value step by step:

1. Find the probability of rolling a 6: 1/6.
2. Multiply the probability by the value of rolling a 6: (1/6) * $3.00 = $0.50.
3. Find the probability of not rolling a 6: 1 - 1/6 = 5/6.
4. Multiply the probability by the cost of each roll: (5/6) * (-$0.90) = -$0.75.
(Note: We use a negative value for the cost since it is an expense.)
5. Add the two values together: $0.50 + (-$0.75) = -$0.25.

The expected value of this game is -$0.25.

Explanation: To calculate the expected value, we consider the probability of each possible outcome and multiply it by its corresponding value. In this case, we find the probability of rolling a 6 (1/6) and multiply it by the value of winning ($3.00). We then find the probability of not rolling a 6 (5/6) and multiply it by the cost of each roll (-$0.90). Adding these products together gives us the expected value of the game, which is -$0.25.