Question

If you toss a pair of dice, the sum \(T\) of the numbers appearing on the upper faces of the dice can assume the value of an integer in the interval \(2 \leq T \leq 12\) a. Find the probability distribution for \(T\). Display this probability distribution in a table. b. Construct a probability histogram for \(P(T)\). How would you describe the shape of this distribution?

Short answer

Answer: The probability of rolling a sum of 7 with two dice is 6/36 or 1/6. The shape of the probability distribution for the sum of two dice is symmetric, with a peak around the value of 7, meaning that the sum of 7 is the most likely outcome and the probabilities decrease as we move away from 7 in both directions.

Step-by-step solution

Understand the problem and list out all possible outcomes

First, we need to find all possible outcomes when two dice are rolled. There are 6 faces on each die, so there are a total of 6 x 6 = 36 possible outcomes when two dice are rolled.

Count occurrences of each sum T

Next, we need to calculate how many times each sum from 2 to 12 appears among the possible outcomes. We will create a table to record these values: T (sum) | Frequency ------------|-------------- 2 | 1 3 | 2 4 | 3 5 | 4 6 | 5 7 | 6 8 | 5 9 | 4 10 | 3 11 | 2 12 | 1 For example, the sum T = 2 can only occur with the outcome (1,1); T = 3 can occur with (1,2) and (2,1), and so on.

Calculate probabilities for each sum T

With the frequencies of each sum calculated, we can now find the probabilities for each sum by dividing the frequency by the total number of possible outcomes (36). Add these probabilities to the table: T (sum) | Frequency | Probability ------------|--------------|----- 2 | 1 | 1/36 3 | 2 | 2/36 4 | 3 | 3/36 5 | 4 | 4/36 6 | 5 | 5/36 7 | 6 | 6/36 8 | 5 | 5/36 9 | 4 | 4/36 10 | 3 | 3/36 11 | 2 | 2/36 12 | 1 | 1/36

Construct a probability histogram

Now that we have found the probability distribution for T, we need to represent it visually in the form of a histogram. To construct the histogram, use the sum T on the x-axis and the corresponding probability on the y-axis. Create bars for each sum T from 2 to 12, with height proportional to the probability.

Describe the shape of the distribution

By observing the probability histogram, we can describe its shape. The histogram of the sum of two dice is symmetric and has a peak around the value of 7. This means that the sum of 7 is the most likely outcome, and as we move away from 7 (in either direction), the probability of the sum occurring decreases.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. When you roll a pair of dice, probability helps you understand how often each possible sum of dice might occur. In probability theory, an event is an outcome or a set of outcomes. In the case of two dice, your possible events include sums like 2, 3, 4, up to 12.

Each possible result has a corresponding probability, which is the ratio of favorable outcomes to the total number of outcomes. Here, since each die has 6 sides, two dice have a combined 6 x 6 = 36 possible outcomes. The probability of each sum is given by dividing its number of occurrences by 36.

Understanding these probabilities helps in predicting how often a particular sum might appear in practice. By organizing these predictions into a probability distribution, you can harness this information for planning and decision-making. Probability theory provides tools for making sense of random phenomena and quantifying uncertainty.

Discrete Random Variables

In probability theory, a discrete random variable is a type of random variable that can take on a countable number of distinct values. The sums of a pair of dice, for example, are considered discrete random variables. They can only take values within a specific set - in this case, integers between 2 and 12.

Whenever you roll a pair of dice, the result (the sum of the numbers on the upper faces) lends itself to being modeled as a discrete random variable. Each possible sum corresponds to a specific value that the random variable, let's call it \( T \), can assume.

This concept becomes crucial when you calculate the probability of each sum occurring. Since you are dealing with countable and finite outcomes, you can easily determine the probability distribution by counting the occurrences of each possible sum. These values, and their probabilities, are then organized into a probability distribution that helps visualize which sums are most and least likely to occur.

Histogram

To visually represent a probability distribution, a histogram can be used. A histogram is a graphical representation that organizes a group of data points into specified ranges. It is an effective way to summarize large datasets and see patterns easily.

In the context of rolling two dice, a probability histogram aids in showing the frequency distribution of the sums. On the x-axis, you place the possible sums, ranging from 2 to 12. On the y-axis, you present the probabilities associated with each sum.

Each bar represents the probability of each sum, with taller bars indicating more likely sums. The histogram provides an immediate visual understanding of which sums are common (like 7) and which are rare (like 2 and 12). This method is particularly useful for communicating insights gained from the probability distribution in an intuitive manner.

Symmetry in Probability Distributions

When we talk about symmetry in probability distributions, we're referring to how the distribution's values are balanced around a central point. For the sum of two dice, the probability distribution is symmetric around the sum of 7.

Symmetrical distributions have the same shape on both sides of a central point, meaning the probabilities are mirrored. In our dice example, the probabilities rise as you approach 7 and decrease similarly as you move away from 7 in both directions.

This symmetry suggests some interesting properties, like the sum of 7 being most frequent, which is intuitive since there are more combinations that total 7 than any other number. Practically, this symmetrical property can be used in various applications, like enhancing strategies in games involving dice, or predicting outcomes over a large number of trials.