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Chapter 7: Problem 8

What is the expected sum of the numbers that appear when three fair dice are rolled?

Short Answer

Expert verified
10.5

Step by step solution

01

Determine the expected value for one die

Each side of a fair six-sided die (1, 2, 3, 4, 5, 6) has an equal probability of \(\frac{1}{6}\). The expected value (E) of a die can be calculated using the formula for the expected value of a uniform distribution:\[ E = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 \]
02

Calculate the expected sum for three dice

The expected sum when multiple dice are rolled is the sum of the expected values of each individual die. Since each die has an expected value of 3.5, and there are three dice, the calculation is:\[ E(\text{sum}) = 3.5 + 3.5 + 3.5 = 3.5 \times 3 = 10.5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a way to measure the likelihood of a certain event happening. It ranges from 0 to 1, where 0 means the event cannot happen, and 1 means the event will certainly happen. In the context of rolling a die, each side has an equal chance of appearing. Since a fair six-sided die has six faces, each face has a probability of \( \frac{1}{6} \). When we roll three dice, we consider the probability for each of the dice to contribute to the result independently.
Expected Value
The expected value (E) is a fundamental concept in probability and statistics. It represents the average outcome of a random event if the same experiment is repeated many times. For a single six-sided die, the expected value is calculated by adding the values of all possible outcomes and dividing by the number of outcomes: \[ E = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 \]. This means that, on average, you can expect a roll of a die to result in 3.5 over a large number of trials.
Uniform Distribution
A uniform distribution is a type of probability distribution where all outcomes are equally likely. In the case of a fair six-sided die, each number from 1 to 6 is equally likely to occur, representing a uniform distribution. This is why each number has a probability of \( \frac{1}{6} \). This equal likelihood simplifies the calculation of the expected value and ensures that each face contributes equally to the average outcome.
Sum of Random Variables
When dealing with multiple dice, the sum of their outcomes is a random variable in itself. The expected value of the sum of independent random variables is the sum of their expected values. Therefore, if each die has an expected value of 3.5, the expected sum for three dice is: \[ E(\text{sum}) = 3.5 + 3.5 + 3.5 = 10.5 \]. This rule holds generally and is extremely useful in various applications of probability and statistics, making calculations straightforward when dealing with sums of independent events.

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Most popular questions from this chapter

A pair of dice is rolled in a remote location and when you ask an honest observer whether at least one die came up six, this honest observer answers in the affirmative. a) What is the probability that the sum of the numbers that came up on the two dice is seven, given the information provided by the honest observer? b) Suppose that the honest observer tells us that at least one die came up five. What is the probability the sum of the numbers that came up on the dice is seven, given this information?

Use Chebyshev's inequality to find an upper bound on the probability that the number of tails that come up when a biased coin with probability of heads equal to 0.6 is tossed \(n\) times deviates from the mean by more than \(\sqrt{n}\) .

A group of six people play the game of 鈥渙dd person out鈥� to determine who will buy refreshments. Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the probability that there is an odd person out after the coins are flipped once?

A pair of dice is loaded. The probability that a 4 appears on the first die is \(2 / 7,\) and the probability that a 3 appears on the second die is \(2 / 7 .\) Other outcomes for each die appear with probability \(1 / 7 .\) What is the probability of 7 appearing as the sum of the numbers when the two dice are rolled?

Show that if \(E\) and \(F\) are independent events, then \(\overline{E}\) and \(\overline{F}\) are also independent events.
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