Assume That A Fair Die Is Rolled. The Sample Space Is $\{1,2,3,4,5,6\}$, And All The Outcomes Are Equally Likely. Find $P(\text{Multiple Of } 4$\]. Express Your Answer In Exact Form.$P(\text{Multiple Of } 4) =$ $\square$

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Introduction

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When a fair die is rolled, there are six possible outcomes, each equally likely. The sample space is given by $\{1,2,3,4,5,6\}$. In this article, we will find the probability of rolling a multiple of 4 on a fair die.

Understanding the Sample Space

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The sample space is the set of all possible outcomes of rolling a fair die. In this case, the sample space is $\{1,2,3,4,5,6\}$. Each outcome is equally likely, meaning that the probability of each outcome is $\frac{1}{6}$.

Defining the Event

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The event we are interested in is rolling a multiple of 4. This means that the outcome must be either 4 or 12 (although 12 is not a possible outcome on a standard six-sided die).

Counting the Favorable Outcomes

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The favorable outcomes are the outcomes that satisfy the event. In this case, the favorable outcomes are $\{4\}$, since 4 is the only multiple of 4 that can be rolled on a standard six-sided die.

Calculating the Probability

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The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, the number of favorable outcomes is 1 (since there is only one multiple of 4 that can be rolled), and the total number of possible outcomes is 6.

Therefore, the probability of rolling a multiple of 4 on a fair die is:

$$P(\text{Multiple of } 4) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{6}$$

Conclusion

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In conclusion, the probability of rolling a multiple of 4 on a fair die is $\frac{1}{6}$. This means that if a fair die is rolled many times, we would expect to see a multiple of 4 about $\frac{1}{6}$ of the time.

Example

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Suppose we roll a fair die 36 times. How many times would we expect to see a multiple of 4?

Using the probability we calculated earlier, we can multiply the probability by the number of trials to get the expected number of successes:

$$\text{Expected number of successes} = P(\text{Multiple of } 4) \times \text{Number of trials} = \frac{1}{6} \times 36 = 6$$

Therefore, we would expect to see a multiple of 4 about 6 times in 36 rolls.

Simplifying the Fraction

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The fraction $\frac{1}{6}$ is already in its simplest form, so there is no need to simplify it further.

Converting to Decimal

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If we want to express the probability as a decimal, we can divide the numerator by the denominator:

$$P(\text{Multiple of } 4) = \frac{1}{6} = 0.1667$$

However, it's generally more accurate to leave the probability as a fraction, since fractions can be expressed exactly, whereas decimals may be rounded or truncated.

Using the Probability in Real-World Situations

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The probability of rolling a multiple of 4 on a fair die can be used in a variety of real-world situations. For example, suppose we are playing a game that involves rolling a die, and we want to know the probability of rolling a multiple of 4. We can use the probability we calculated earlier to answer this question.

Alternatively, suppose we are conducting an experiment that involves rolling a die many times, and we want to know the expected number of times we will see a multiple of 4. We can use the probability we calculated earlier to answer this question.

Conclusion

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In conclusion, the probability of rolling a multiple of 4 on a fair die is $\frac{1}{6}$. This probability can be used in a variety of real-world situations, and it can be expressed exactly as a fraction.

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Q: What is the probability of rolling a multiple of 4 on a fair die?

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A: The probability of rolling a multiple of 4 on a fair die is $\frac{1}{6}$. This means that if a fair die is rolled many times, we would expect to see a multiple of 4 about $\frac{1}{6}$ of the time.

Q: How do you calculate the probability of rolling a multiple of 4 on a fair die?

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A: To calculate the probability of rolling a multiple of 4 on a fair die, we need to count the number of favorable outcomes (i.e., the outcomes that satisfy the event) and divide it by the total number of possible outcomes. In this case, the number of favorable outcomes is 1 (since there is only one multiple of 4 that can be rolled), and the total number of possible outcomes is 6.

Q: What is the expected number of times we will see a multiple of 4 in 36 rolls of a fair die?

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A: Using the probability we calculated earlier, we can multiply the probability by the number of trials to get the expected number of successes:

$$\text{Expected number of successes} = P(\text{Multiple of } 4) \times \text{Number of trials} = \frac{1}{6} \times 36 = 6$$

Therefore, we would expect to see a multiple of 4 about 6 times in 36 rolls.

Q: Can we simplify the fraction $\frac{1}{6}$?

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A: The fraction $\frac{1}{6}$ is already in its simplest form, so there is no need to simplify it further.

Q: How do you convert the fraction $\frac{1}{6}$ to a decimal?

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A: If we want to express the probability as a decimal, we can divide the numerator by the denominator:

$$P(\text{Multiple of } 4) = \frac{1}{6} = 0.1667$$

However, it's generally more accurate to leave the probability as a fraction, since fractions can be expressed exactly, whereas decimals may be rounded or truncated.

Q: What are some real-world situations where we might use the probability of rolling a multiple of 4 on a fair die?

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A: The probability of rolling a multiple of 4 on a fair die can be used in a variety of real-world situations. For example, suppose we are playing a game that involves rolling a die, and we want to know the probability of rolling a multiple of 4. We can use the probability we calculated earlier to answer this question.

Alternatively, suppose we are conducting an experiment that involves rolling a die many times, and we want to know the expected number of times we will see a multiple of 4. We can use the probability we calculated earlier to answer this question.

Q: Can we use the probability of rolling a multiple of 4 on a fair die to make predictions about future events?

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A: Yes, we can use the probability of rolling a multiple of 4 on a fair die to make predictions about future events. For example, if we roll a fair die many times, we would expect to see a multiple of 4 about $\frac{1}{6}$ of the time. This means that if we roll a fair die 36 times, we would expect to see a multiple of 4 about 6 times.

Q: What are some common mistakes people make when calculating the probability of rolling a multiple of 4 on a fair die?

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A: Some common mistakes people make when calculating the probability of rolling a multiple of 4 on a fair die include:

• Not counting the number of favorable outcomes correctly
• Not dividing the number of favorable outcomes by the total number of possible outcomes
• Not expressing the probability as a fraction or decimal
• Not using the correct probability in real-world situations

Q: How can we use the probability of rolling a multiple of 4 on a fair die to make informed decisions?

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A: We can use the probability of rolling a multiple of 4 on a fair die to make informed decisions by considering the probability of different outcomes and making decisions based on that probability. For example, if we are playing a game that involves rolling a die, and we want to know the probability of rolling a multiple of 4, we can use the probability we calculated earlier to make informed decisions about our strategy.

Q: Can we use the probability of rolling a multiple of 4 on a fair die to make predictions about events that are not related to rolling a die?

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A: No, we cannot use the probability of rolling a multiple of 4 on a fair die to make predictions about events that are not related to rolling a die. The probability of rolling a multiple of 4 on a fair die is a specific probability that is only applicable to the event of rolling a die, and it cannot be used to make predictions about other events.