Problem 25 - Precalculus with Limits

Question

In Exercises 25 - 30, find the probability for the experiment of tossing a six-sided die twice. The sum is \( 6 \).

Step-by-Step Solution

Answer

The probability of getting a sum of 6 when tossing a six-sided die twice is \( \frac{5}{36} \).

Step 1: Identify all possible outcomes

We first identify all possible outcomes when the die is tossed twice. This can be done by enumerating all the possibilities in a table. This results in \(6 \times 6 = 36\) possible outcomes in total.

Step 2: Identify favorable outcomes

A favorable outcome refers to an outcome in which the sum is 6. By looking at the outcome table, we can see that there are five combinations that can add up to 6. These combinations are (1,5), (2,4), (3,3), (4,2), and (5,1). Each combination is a different outcome, so there are five favorable outcomes in total.

Step 3: Calculate the probability

The probability is calculated by dividing the number of favorable outcomes by the number of total outcomes. The formula for probability is \( P(A) = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of outcomes}}}} \) . In this case, the number of favorable outcomes is 5, and the total number of outcomes is 36. So, \( P(A) = \frac{5}{36} \)

Key Concepts

Understanding Favorable Outcomes

When discussing probability, a key concept is that of "favorable outcomes." This term refers to the specific results that we're interested in when conducting an experiment or assessing a scenario.
A favorable outcome is not necessarily a good or preferable outcome in the general sense, but rather just the one that meets the criteria set by a question or situation.
In the context of our original problem, where we want the sum of the faces of a six-sided die tossed twice to be 6, each combination of die rolls that results in this sum is considered favorable.

• The outcome of rolling a 1 and a 5
• The outcome of rolling a 2 and a 4
• The outcome of rolling a 3 and a 3
• The outcome of rolling a 4 and a 2
• The outcome of rolling a 5 and a 1

The total number of favorable outcomes in this scenario is therefore 5, because there are five ways to roll a total sum of 6. Remember, identifying these outcomes accurately is crucial for calculating the probability.

Exploring the Six-Sided Die

A six-sided die is a cube with numbers marked on each face, from 1 to 6. Each number represents a possible outcome when the die is rolled. The die is very balanced, making each outcome equally possible.
In probability, when you roll a six-sided die once, there are 6 possible outcomes, corresponding to each of the faces: 1, 2, 3, 4, 5, or 6.
When we involve the die in a probabilistic experiment like the one described, the balance ensures that every outcome has an equal chance of occurring, which simplifies probability calculations.
When tossing a six-sided die twice, like in our exercise, we multiply the outcomes:

• The first roll results in any of the 6 numbers.
• The second roll, independent of the first, also results in any of the 6 numbers.

Thus, the total number of combined outcomes for two dice rolls is 6 multiplied by 6, resulting in 36 possible outcomes. This comprehensive enumeration allows us to consider every possible result when calculating probabilities.

Identifying Possible Outcomes

Possible outcomes refer to all the potential results that can occur during an experiment. In probability, identifying all possible outcomes is a crucial first step in any analysis.
When using a six-sided die twice, there are 6 outcomes for the first roll and another 6 for the second.
By multiplying these, we see that the total number of possible outcomes for the experiment is 36.

• The first die's result could be any number from 1 to 6.
• The second die also has 6 options (1 through 6) for each result of the first die.
• Hence, pair each first roll with each second one, creating 36 unique outcomes.

These possible outcomes form the basis for calculating probability. Recognizing each combination (for instance, (1,1), (1,2), ..., through (6,6)) ensures that we don't overlook any possibilities. Once all possible outcomes are identified, we can accurately count how many meet our criteria for being favorable, like those equaling a sum of 6.