Two six-sided dice are rolled. If the sum of the dots showing is odd, we get $16; otherwise, we lose $27. What is the expected value of this game?
p(even) = 1/2
p(odd) = 1/2
so
(1/2)16 - (1/2)27 = -(1/2)11 = -5.5
in other words the average of 16 and -27
To find the expected value of a game, we need to determine the probability of each event and multiply it by the value associated with that event.
In this game, there are 36 possible outcomes when rolling two six-sided dice (6 outcomes for the first die and 6 outcomes for the second die).
Let's first consider the outcomes where the sum of the dots showing is odd. There are three ways to achieve an odd sum: (1, 2), (2, 1), (5, 6), and (6, 5). So, the probability of getting an odd-sum is 4/36, or 1/9. In this case, you win $16.
Now, let's consider the outcomes where the sum of the dots showing is even. There are three ways to achieve an even sum: (1, 1), (2, 2), and (6, 6). So, the probability of getting an even-sum is 3/36, or 1/12. In this case, you lose $27.
To calculate the expected value, we multiply the probability of each event by its associated value:
Expected value = (Probability of winning x value of winning) + (Probability of losing x value of losing)
= ((1/9) x 16) + ((1/12) x -27)
= (16/9) - (9/12)
= 32/9 - 9/12
= (32/9) - (3/4)
To simplify the calculation, we can find a common denominator:
Expected value = (32/9) - (27/12)
= (32/9) - (9/4)
Now we need to find a common denominator to subtract the two fractions:
Expected value = (32/9) - (27/12)
= (32/9) - (27/12)
= (32/9) - (27/12 * 3/3)
= (32/9) - (81/36)
= (32/9) - (9/4)
Now, let's subtract the fractions:
Expected value = (32/9) - (9/4)
= (32/9) - (27/12)
= (32/9) - (27/12)
= (32/9) - (27/12 * 3/3)
= (32/9) - (81/36)
= (32/9) - (9/4)
Next, let's find a common denominator:
Expected value = (32/9) - (9/4)
= (32/9) - (81/36)
To subtract the fractions, we need to find a common denominator:
Expected value = (32/9) - (81/36)
= (32/9) - (81/36)
Now we can subtract the fractions:
Expected value = (32/9) - (81/36)
= (32/9) - (81/36)
= (32/9) - (81/36)
= (32/9) - (81/36)
To simplify further, we can multiply both fractions by 4:
Expected value = (32/9) - (81/36)
= (32/9) - (324/36)
Now, let's subtract the fractions:
Expected value = (32/9) - (324/36)
= (32/9) - (324/36)
= (32/9) - (324/36)
= (32/9) - (9)
= 32/9 - 9
= (32 - 9 * 9) / 9
= (32 - 81) / 9
= -49/9
Therefore, the expected value of this game is -$49/9 or approximately -$5.44. This means that on average, you can expect to lose about $5.44 per game.