Two six-sided dice are rolled. If the sum of the dots showing is odd, we get $10; otherwise, we lose $25. What is the expected value of this game?
ur rong
If you look at the outcomes, you will see that half of the sums are even, the other half are odd.
So prob (even) = 1/2
prob(odd) = 1/2
expected value= (1/2)$10 + (1/2(-$25) = -7.5
Who would possible want to play this game ?
To find the expected value of this game, we need to calculate the probability of each outcome and multiply it by the corresponding payoff.
There are a total of 36 possible outcomes when rolling two six-sided dice (each die has 6 possible outcomes).
Let's first calculate the probability of getting an odd sum:
The possible odd sums are: 1, 3, 5, 7, 9, 11
To determine the probability of each odd sum, we can use a table showing all the combinations of the dice rolls:
Roll 1 | Roll 2
------|------
1 | 1
1 | 2
1 | 3
1 | 4
1 | 5
1 | 6
2 | 1
2 | 2
2 | 3
2 | 4
2 | 5
2 | 6
3 | 1
3 | 2
3 | 3
3 | 4
3 | 5
3 | 6
4 | 1
4 | 2
4 | 3
4 | 4
4 | 5
4 | 6
5 | 1
5 | 2
5 | 3
5 | 4
5 | 5
5 | 6
6 | 1
6 | 2
6 | 3
6 | 4
6 | 5
6 | 6
Out of the 36 possible outcomes:
- There are 18 outcomes with an odd sum (1, 3, 5, 7, 9, 11).
- There are 18 outcomes with an even sum (2, 4, 6, 8, 10, 12).
Thus, the probability of getting an odd sum is 18/36 = 1/2.
Now, let's calculate the expected value:
If the sum is odd, we get $10. So, the payoff in this case is +$10.
If the sum is even, we lose $25. So, the payoff in this case is -$25.
The expected value is calculated as follows:
Expected Value = (Probability of odd sum × Payoff for odd sum) + (Probability of even sum × Payoff for even sum)
Expected Value = (1/2 × $10) + (1/2 × -$25) = $5 - $12.5 = -$7.5
Therefore, the expected value of this game is -$7.5.
To find the expected value of this game, we need to calculate the probability of each outcome and multiply it by the corresponding payout.
Let's first look at the possible outcomes of rolling two six-sided dice. There are 36 possible outcomes because each die has 6 possible results, and we multiply the number of outcomes of one die by the number of outcomes of the other (6x6=36).
Now, let's consider the possible sums and their corresponding probabilities:
Odd Sums:
To get an odd sum, one die must show an even number and the other die must show an odd number. There are 3 even numbers (2, 4, 6) and 3 odd numbers (1, 3, 5) on a standard six-sided die. So, the probability of getting an odd sum is (3/6) x (3/6) = 9/36 = 1/4.
Payout for an odd sum: $10.
Even Sums:
To get an even sum, both dice must either show an even number or both must show an odd number. There are 3 even numbers and 3 odd numbers, so the probability of getting an even sum is (3/6) x (3/6) + (3/6) x (3/6) = 18/36 = 1/2.
Payout for an even sum: -$25.
Now, let's calculate the expected value:
Expected value = (Payout for odd sum x Probability of odd sum) + (Payout for even sum x Probability of even sum)
Expected value = ($10 x 1/4) + (-$25 x 1/2) = $2.50 - $12.50 = -$10.
So, the expected value of this game is -$10, meaning that on average, you will lose $10 for each game.