A fair die is rolled 16 times. What is the expected sum of the 16 rolls?
To find the expected sum of the 16 rolls of a fair die, we need to determine the expected value of a single roll and then multiply it by 16 since we are rolling the die 16 times.
In this case, the expected value of a single roll of a fair die can be found by taking the average of the possible outcomes, which range from 1 to 6. The formula to calculate the expected value is:
Expected value = (Sum of all possible outcomes) / (Number of possible outcomes)
For a fair die, the sum of all possible outcomes is 1 + 2 + 3 + 4 + 5 + 6 = 21, and the number of possible outcomes is 6.
So the expected value of a single roll is 21 / 6 = 3.5.
To find the expected sum of the 16 rolls, we multiply the expected value of a single roll by the number of rolls:
Expected sum = Expected value of a single roll * Number of rolls
Expected sum = 3.5 * 16 = 56
Therefore, the expected sum of the 16 rolls is 56.
To find the expected sum of the 16 rolls of a fair die, we need to determine the expected value of a single roll and then multiply it by the number of rolls.
The expected value of a single roll of a fair die is calculated by taking the average of all possible outcomes, each weighted by their respective probabilities. In this case, as the die is fair, there are 6 equally likely outcomes, with each outcome having a probability of 1/6.
The possible outcomes of a single roll are {1, 2, 3, 4, 5, 6}, and their respective probabilities are 1/6, 1/6, 1/6, 1/6, 1/6, 1/6.
To calculate the expected value of a single roll, we sum up the products of each outcome multiplied by its probability:
Expected value of a single roll = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6
Simplifying this expression:
Expected value of a single roll = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
Therefore, the expected sum of the 16 rolls can be calculated by multiplying the expected value of a single roll by the number of rolls:
Expected sum of the 16 rolls = 3.5 * 16 = 56