Calculating the Possible Outcomes

Oct 13, 2025 by vinooalluri.com 120 views

Understanding the Problem: A Relative-Frequency Table for Two Fair, Four-Sided Dice

When working with probability and statistics, it's essential to understand how to calculate and represent the likelihood of different outcomes. In this article, we'll explore the concept of a relative-frequency table and apply it to a specific problem involving two fair, four-sided dice.

You roll two fair, four-sided dice. Let the random variable $X$ represent the sum on the faces of the dice. We need to determine the correct relative-frequency table for the possible values of $X$ .

Calculating the Possible Outcomes

To calculate the possible outcomes, we need to consider the combinations of the two dice. Each die has four possible outcomes (1, 2, 3, or 4), so the total number of possible outcomes for the two dice is $4 \times 4 = 16$ .

Determining the Possible Values of $X$

The possible values of $X$ are the sums of the two dice. We can calculate these values as follows:

$X = 2$ : This occurs when both dice show 1.
$X = 3$ : This occurs when one die shows 1 and the other die shows 2, or when one die shows 2 and the other die shows 1.
$X = 4$ : This occurs when both dice show 2.
$X = 5$ : This occurs when one die shows 1 and the other die shows 4, or when one die shows 4 and the other die shows 1, or when one die shows 2 and the other die shows 3, or when one die shows 3 and the other die shows 2.
$X = 6$ : This occurs when both dice show 3.
$X = 7$ : This occurs when one die shows 1 and the other die shows 6, or when one die shows 6 and the other die shows 1, or when one die shows 2 and the other die shows 5, or when one die shows 5 and the other die shows 2, or when one die shows 3 and the other die shows 4, or when one die shows 4 and the other die shows 3.
$X = 8$ : This occurs when both dice show 4.
$X = 9$ : This occurs when one die shows 3 and the other die shows 6, or when one die shows 6 and the other die shows 3, or when one die shows 4 and the other die shows 5, or when one die shows 5 and the other die shows 4.
$X = 10$ : This occurs when both dice show 5.
$X = 11$ : This occurs when one die shows 5 and the other die shows 6, or when one die shows 6 and the other die shows 5.
$X = 12$ : This occurs when both dice show 6.

Calculating the Probabilities

To calculate the probabilities, we need to count the number of ways each value of $X$ can occur and divide it by the total number of possible outcomes (16).

$P(X = 2) = \frac{1}{16}$
$P(X = 3) = \frac{2}{16} = \frac{1}{8}$
$P(X = 4) = \frac{1}{16}$
$P(X = 5) = \frac{4}{16} = \frac{1}{4}$
$P(X = 6) = \frac{2}{16} = \frac{1}{8}$
$P(X = 7) = \frac{6}{16} = \frac{3}{8}$
$P(X = 8) = \frac{1}{16}$
$P(X = 9) = \frac{4}{16} = \frac{1}{4}$
$P(X = 10) = \frac{1}{16}$
$P(X = 11) = \frac{2}{16} = \frac{1}{8}$
$P(X = 12) = \frac{1}{16}$

The Correct Relative-Frequency Table

Based on the calculations above, the correct relative-frequency table is:

$x$	$P(X=x)$
2	$\frac{1}{16}$
3	$\frac{1}{8}$
4	$\frac{1}{16}$
5	$\frac{1}{4}$
6	$\frac{1}{8}$
7	$\frac{3}{8}$
8	$\frac{1}{16}$
9	$\frac{1}{4}$
10	$\frac{1}{16}$
11	$\frac{1}{8}$
12	$\frac{1}{16}$

In this article, we calculated the possible outcomes and values of $X$ for two fair, four-sided dice. We then determined the probabilities of each value of $X$ and constructed the correct relative-frequency table. This table provides a useful representation of the likelihood of different sums when rolling two fair, four-sided dice.
Frequently Asked Questions: Relative-Frequency Tables for Two Fair, Four-Sided Dice

Q: What is a relative-frequency table?

A: A relative-frequency table is a table that shows the probability of each possible outcome in a random experiment. In the case of two fair, four-sided dice, the relative-frequency table shows the probability of each possible sum.

Q: How do I calculate the probabilities in a relative-frequency table?

A: To calculate the probabilities, you need to count the number of ways each value of $X$ can occur and divide it by the total number of possible outcomes. In the case of two fair, four-sided dice, there are 16 possible outcomes, so you need to count the number of ways each value of $X$ can occur and divide it by 16.

Q: What is the difference between a relative-frequency table and a probability distribution?

A: A relative-frequency table and a probability distribution are similar, but they are not exactly the same thing. A relative-frequency table shows the probability of each possible outcome, while a probability distribution shows the probability of each possible value of a random variable. In the case of two fair, four-sided dice, the relative-frequency table shows the probability of each possible sum, while the probability distribution shows the probability of each possible value of $X$ .

Q: How do I use a relative-frequency table to make predictions about the outcome of a random experiment?

A: To use a relative-frequency table to make predictions about the outcome of a random experiment, you need to look at the probabilities of each possible outcome and use them to make a prediction. For example, if you are rolling two fair, four-sided dice and you want to know the probability of getting a sum of 7, you can look at the relative-frequency table and see that the probability of getting a sum of 7 is $\frac{3}{8}$ .

Q: Can I use a relative-frequency table to make predictions about the outcome of a random experiment if I don't know the total number of possible outcomes?

A: No, you cannot use a relative-frequency table to make predictions about the outcome of a random experiment if you don't know the total number of possible outcomes. The relative-frequency table is based on the total number of possible outcomes, so if you don't know the total number of possible outcomes, you cannot use the relative-frequency table to make predictions.

Q: How do I construct a relative-frequency table for a random experiment with more than two possible outcomes?

A: To construct a relative-frequency table for a random experiment with more than two possible outcomes, you need to count the number of ways each value of $X$ can occur and divide it by the total number of possible outcomes. For example, if you are rolling three fair, six-sided dice and you want to construct a relative-frequency table for the possible sums, you need to count the number of ways each value of $X$ can occur and divide it by the total number of possible outcomes (216).

Q: Can I use a relative-frequency table to make predictions about the outcome of a random experiment if the experiment has more than two possible outcomes?

A: Yes, you can use a relative-frequency table to make predictions about the outcome of a random experiment if the experiment has more than two possible outcomes. The relative-frequency table shows the probability of each possible outcome, so you can use it to make predictions about the outcome of the experiment.

Q: How do I use a relative-frequency table to make predictions about the outcome of a random experiment with more than two possible outcomes?

A: To use a relative-frequency table to make predictions about the outcome of a random experiment with more than two possible outcomes, you need to look at the probabilities of each possible outcome and use them to make a prediction. For example, if you are rolling three fair, six-sided dice and you want to know the probability of getting a sum of 10, you can look at the relative-frequency table and see that the probability of getting a sum of 10 is $\frac{27}{216}$ .

In this article, we have answered some frequently asked questions about relative-frequency tables for two fair, four-sided dice. We have discussed how to calculate the probabilities in a relative-frequency table, how to use a relative-frequency table to make predictions about the outcome of a random experiment, and how to construct a relative-frequency table for a random experiment with more than two possible outcomes. We hope that this article has been helpful in answering your questions about relative-frequency tables.