Problem 42 - College Algebra

Question

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling any sum other than 5 or 6.

Step-by-Step Solution

Answer

The probability of not rolling a sum of 5 or 6 is \( \frac{3}{4} \).

Step 1: Define the Total Sample Space

Two dice are rolled and each die has 6 faces. The total number of possible outcomes when rolling two dice is given by multiplying the number of outcomes for each die: \(6 \times 6 = 36\). This means there are 36 different outcomes when two dice are rolled.

Step 2: List Outcomes for Sums of 5 or 6

We need to find out the outcomes that lead to sums of 5 or 6:- Sum of 5: Possible combinations are (1,4), (2,3), (3,2), and (4,1). That gives us 4 outcomes.- Sum of 6: Possible combinations are (1,5), (2,4), (3,3), (4,2), and (5,1). This gives us 5 outcomes.So, there are a total of \(4 + 5 = 9\) outcomes for sums of 5 or 6.

Step 3: Calculate Outcomes Excluding Sums of 5 or 6

To find the outcomes for any sum other than 5 or 6, we subtract the outcomes of sums 5 and 6 from the total outcomes.Total outcomes excluding sums of 5 or 6: \(36 - 9 = 27\).

Step 4: Calculate Probability of Desired Event

The probability of an event is given by \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \). So, the probability of rolling any sum other than 5 or 6 is \( \frac{27}{36} \).

Step 5: Simplify the Probability Expression

To simplify \( \frac{27}{36} \), we find the greatest common divisor (GCD) of 27 and 36, which is 9. Dividing both numerator and denominator by 9, we get:\[ \frac{27}{36} = \frac{27 \div 9}{36 \div 9} = \frac{3}{4} \].

Key Concepts

Sample Space

When dealing with probability, understanding the concept of sample space is crucial. The sample space includes all possible outcomes of an experiment.

In the case of rolling two dice, each die has 6 sides, labeled from 1 to 6. To find the sample space, multiply the number of outcomes for one die by the number of outcomes for the other die.

• This means you calculate: \(6 \times 6 = 36\).
• There are 36 different possibilities when rolling two dice.

Each combination represents a separate outcome, such as rolling a 1 on the first die and a 3 on the second die, which is noted as \((1,3)\). The entire list of outcomes forms our sample space, an essential foundation for solving probability problems.

Dice Combinations

Understanding dice combinations helps us identify the various ways numbers can be summed when rolling two dice. To solve for specific sums, it's useful to list out which dice combinations result in those numbers.

For instance, to find the dice combinations that add up to 5:

• (1, 4)
• (2, 3)
• (3, 2)
• (4, 1)

This gives us a total of 4 outcomes for a sum of 5.

For a sum of 6, the possible combinations are:

• (1, 5)
• (2, 4)
• (3, 3)
• (4, 2)
• (5, 1)

And there are 5 outcomes for a sum of 6.

Knowing these combinations is key to calculating precise probabilities for specific events.

Favorable Outcomes

When calculating probabilities, identifying the favorable outcomes is essential. Favorable outcomes are those outcomes that meet the specific conditions of the problem you're trying to solve. In this scenario, we want results excluding sums of 5 or 6.

First, we determine all possible sums that are not 5 or 6. There are 36 total outcomes when two dice are rolled. We already calculated that there are 9 outcomes which yield a 5 or 6.

By subtracting the number of these specific outcomes from the total, we find the number of favorable outcomes:

• Total outcomes for sums 5 or 6: 9
• Total possible outcomes: 36
• Favorable outcomes: \(36 - 9 = 27\)

This means there are 27 outcomes where the sum isn't 5 or 6, and these are the favorable outcomes we consider when calculating the probability.

Simplifying Fractions

Simplifying fractions is a crucial step in making probability results clearer and simpler. After determining the favorable outcomes, the probability expression often results in a fraction that can be simplified.

For example, the probability of rolling a sum other than 5 or 6 is initially calculated as \(\frac{27}{36}\). To simplify this fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator. For 27 and 36, the GCD is 9.
• Divide both the numerator and the denominator by the GCD.
• This leads to: \[\frac{27}{36} = \frac{27 \div 9}{36 \div 9} = \frac{3}{4}\]

Simplifying the fraction allows everyone to easily grasp the probability as \(\frac{3}{4}\), meaning there's a three out of four chance of rolling any sum other than 5 or 6.