Problem 1 - Algebra for College Students

Question

Two dice are tossed. Find the probability of rolling each of the following events: \text { A sum of } 6 \frac{5}{36}

Step-by-Step Solution

Answer

The probability of rolling a sum of 6 is \(\frac{5}{36}\).

Step 1: Identify Possible Outcomes

When two dice are tossed, each die has 6 faces, leading to a total of \(6 \times 6 = 36\) possible outcomes. Each outcome is an ordered pair \((x, y)\), where \(x\) and \(y\) are the numbers on the first and second die, respectively.

Step 2: Define the Desired Event

We are interested in the event where the sum of the numbers on the two dice equals 6. We need to determine which combinations of numbers on the dice lead to this sum.

Step 3: Enumerate Favorable Outcomes

To find combinations that sum to 6: \((1, 5), (2, 4), (3, 3), (4, 2), (5, 1)\). These are the only combinations where the sum of the two dice is 6. Therefore, there are 5 favorable outcomes.

Step 4: Calculate the Probability

The probability of an event is given by dividing the number of favorable outcomes by the total number of possible outcomes. \(P( ext{sum of 6}) = \frac{5}{36}\) because there are 5 favorable outcomes and 36 possible outcomes.

Key Concepts

Favorable Outcomes

When we talk about favorable outcomes in probability, we mean the specific scenarios that match the event we are interested in. In the dice roll problem, the event of interest is getting a sum of 6. So, favorable outcomes are those specific rolls of the dice that result in this sum. For the two dice being rolled:

• The pair (1, 5) has a sum of 6.
• The pair (2, 4) has a sum of 6.
• The pair (3, 3) has a sum of 6.
• The pair (4, 2) has a sum of 6.
• The pair (5, 1) has a sum of 6.

Thus, there are 5 favorable outcomes when trying to achieve a sum of 6 with two dice. It's important to recognize these specific combinations, as they directly influence the probability of the event happening.

Sample Space

In probability, the sample space is the set of all possible outcomes of a random experiment. For our dice example, this means considering every possible way the two dice can land. Each die has 6 sides, resulting in a total of 36 possible combinations, since each die is independent. The pairs are in the form (x, y) where:

• - "x" is the outcome of the first die, and
• - "y" is the outcome of the second die.

To visualize the full sample space, imagine a grid where one axis represents the first die and the other axis represents the second die. Each point on the grid represents a unique outcome, like (1, 1), (1, 2), ..., all the way to (6, 6).
The broader the sample space, the more diverse the outcomes that are considered, but for any dice roll between two six-sided dice, it's always these 36 combinations.

Calculating Probability

To calculate the probability of a particular event, you need to know two things: the number of favorable outcomes and the total number of outcomes in the sample space. The formula for probability is given by:
\[ P( ext{event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]In the dice problem, we identified 5 favorable outcomes that result in a sum of 6. Since the total number of possible outcomes (sample space) when two dice are rolled is 36, we substitute these values into the formula:

• - Number of Favorable Outcomes = 5
• - Total Number of Outcomes = 36

Therefore, the probability of rolling a sum of 6 with two dice is:
\[ P( ext{sum of 6}) = \frac{5}{36} \]Understanding this calculation allows you to predict the likelihood of different events based on their favorable scenarios and the total possible outcomes.