Problem 6 - A First Course in Probability

Question

A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.

Step-by-Step Solution

Answer

The expected sum of 10 rolls of a fair die is 35.

Step 1: Determine the expected value of a single roll

The expected value of a single roll is the average value of all possible outcomes. In the case of a fair six-sided die, the possible outcomes are the numbers 1 through 6, and the probability of each outcome is 1/6. The expected value can be calculated using the following formula: Expected Value = Σ (value × probability) For a fair die, the expected value is: Expected Value = \(1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6})\)

Step 2: Calculate the expected value of a single roll

Using the above formula, we can now calculate the expected value of a single roll: Expected Value = \(1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6})\) Expected Value = \(\frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6}\) Expected Value = \(\frac{21}{6}\) Expected Value = 3.5

Step 3: Calculate the expected sum of 10 rolls

Now that we know the expected value of a single roll (3.5), we can multiply it by the number of rolls (10) to find the expected sum of the 10 rolls: Expected Sum = (Expected Value of a Single Roll) × (Number of Rolls) Expected Sum = 3.5 × 10 = 35 So, the expected sum of 10 rolls of a fair die is 35.

Key Concepts

Probability

The concept of probability revolves around the likelihood of an event occurring. When rolling a fair die, each face has an equal chance of landing face up. This is what we call uniform probability. With a six-sided die, each number from 1 to 6 has a probability of 1/6 of appearing on any given roll.

Probability forms the backbone of understanding expected values. It's essentially the assessment of all possible outcomes and how likely each is to occur.

• If an experiment has multiple outcomes, the probability of each outcome must add up to 1.
• For example, the sum of probabilities for a die’s outcomes 1, 2, 3, 4, 5, and 6 is exactly 1, confirming our calculations are correct.
• Understanding probability helps us make predictions about future events, based on the likelihood of various outcomes.

Random Variables

A random variable is a numerical representation of the outcomes in a probability experiment. In the context of rolling a die, each face represents a different numerical outcome. We treat these possible outcomes as random variables, since they're based on chance.

The concept of random variables helps us quantify the expected value in the exercise.

• Random variables can be categorized as discrete or continuous. A die outcome is a discrete random variable because it has specific, separate values: 1 to 6.
• In the case of our die roll, the values we assign depend solely on probability, where each face of the die represents a potential value the random variable can take.
• The sum of random variables, such as total points from multiple rolls, illustrates how individual random outcomes aggregate to form a larger picture.

Summation

Summation is the process of adding a sequence of numbers, and it is denoted by the symbol \(\Sigma\). In our exercise, summation helps calculate the expected value of rolling a die multiple times.

Summation encompasses the entire set of outcomes of a die roll, allowing us to find the average outcome, also known as the expected value.

• The calculation of an expected value involves multiplying each outcome by its probability and then summing all products together.
• For a single die roll, this is shown in the formula: \(1(\frac{1}{6}) + 2(\frac{1}{6}) +...+ 6(\frac{1}{6}) = \frac{21}{6} = 3.5\).
• When rolling the die 10 times, the summation concept simplifies to multiplying the single roll expected value by the number of rolls, making the computation straightforward: \(3.5 \times 10 = 35\).

Integration of summation with expected values allows us to generalize results over numerous trials, aiding in understanding long-term outcomes of probabilistic events.