Question

Suppose you roll the die 66 times. How many times are you likely to roll a 6? (A) 6 (B) 11 (C) 55 (D) 66

Short answer

The short answer is: You are likely to roll a 6 eleven times if you roll the die 66 times. The calculation is based on the probability of rolling a 6 (\(\frac{1}{6}\)) multiplied by the total number of trials (66): Expected number of 6s = \(\frac{1}{6}\) × 66 = 11.

Step-by-step solution

Find the probability of rolling a 6

To find the probability of rolling a 6, we need to know the total number of possible outcomes and the number of successful outcomes (rolling a 6). There are 6 possible outcomes on a fair die (1, 2, 3, 4, 5, and 6), and only 1 of those is a successful outcome (rolling a 6). So the probability of rolling a 6 on a single roll is \(P(6)=\frac{1}{6}\).

Multiply the probability of rolling a 6 by the total number of trials

In this case, the total number of trials is 66. Since the probability remains constant for each trial, we can calculate the expected number of successful outcomes (rolling a 6) by multiplying the probability of rolling a 6 by the total number of trials: Expected number of 6s = Probability of rolling a 6 × Total number of trials = \(\frac{1}{6}\) × 66.

Calculate the expected number of successful outcomes (rolling a 6)

Now that we have the formula, we can calculate the expected number of successful outcomes (rolling a 6) by performing the multiplication: Expected number of 6s = \(\frac{1}{6}\) × 66 = 11. The answer is (B) 11. You are likely to roll a 6 eleven times if you roll the die 66 times.

Key concepts

Expected Value

Understanding expected value is crucial when dealing with probabilities in repeated experiments or trials. Expected value essentially tells you the average outcome you can expect when you repeat an experiment, like rolling a die, multiple times. It provides a mathematical way to predict the most likely outcome of a given set of trials.

To calculate the expected value, multiply the probability of each outcome by the number of times you perform the experiment. Then, sum these products to arrive at the expected value. For example, if you're rolling a fair six-sided die and want to know how many times you'd expect to roll a "6" in 66 rolls, the expected value can be calculated by multiplying the probability of rolling a 6 by the number of rolls:

• Probability of rolling a 6 = \( \frac{1}{6} \)
• Total rolls = 66
• Expected number of times rolling a 6 = \( \frac{1}{6} \times 66 = 11 \)

Therefore, on average, you can expect to roll a 6 eleven times in 66 rolls.

Rolling a Die

Rolling a die is a classic example of a probability trial with uniform outcomes. A standard six-sided die has equal probability for each of its faces to land on top. Every face of the die (1 through 6) represents one possible outcome, and since all faces are equal, each face has a probability of \( \frac{1}{6} \).

Understanding this uniformity helps when calculating probabilities in various scenarios. Each roll of a die is an independent event, meaning the outcome of one roll doesn't affect the outcome of another. For example, rolling a 6 one time doesn't change the probability that you'll roll another 6 in subsequent rolls.

• Number of faces (outcomes): 6 (1, 2, 3, 4, 5, 6)
• Probability of any single outcome: \( \frac{1}{6} \)

This consistency is what makes dice a perfect tool for illustrating the concepts of probability and expected value.

Successful Outcomes

In probability, "successful outcomes" refers to the specific outcomes you are interested in among all possible outcomes. For instance, if you wish to determine the chance of rolling a 6 with a die, each occurrence of a "6" rolling is considered a successful outcome.

The probability of a successful outcome is the ratio of the number of successful outcomes to the total number of possible outcomes. As illustrated in the die-rolling example, if you want to roll a 6, there is only 1 successful outcome per roll since there is only one face of the die with a 6.

• Number of successful outcomes (rolling a 6): 1
• Total possible outcomes (sides of a die): 6
• Probability of success in one roll: \( \frac{1}{6} \)

By knowing how many times you plan to roll the die, you can calculate how often you expect those successful outcomes to occur by multiplying this probability by the number of trials.