Problem 17 - Algebra for College Students

Question

Solve each problem.Toss a pair of dice. What is the probability of not getting a double? \(\frac{5}{6}\)

Step-by-Step Solution

Answer

The probability of not getting a double is \( \frac{5}{6} \).

Step 1: Identify All Possible Outcomes

When tossing two dice, each die has 6 possible outcomes, resulting in a total of \(6 \times 6 = 36\) possible outcomes. This is the total number of combinations when rolling a pair of dice.

Step 2: Identify Outcome of Getting a Double

A 'double' occurs when both dice show the same number. Thus, the possible doubles are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are 6 such outcomes where the numbers on both dice are the same.

Step 3: Calculate the Probability of Getting a Double

The probability of rolling a double is the number of double outcomes divided by the total number of outcomes. Therefore, it is \( \frac{6}{36} = \frac{1}{6} \).

Step 4: Calculate the Probability of Not Getting a Double

The probability of not rolling a double is the complement of the probability of rolling a double. Hence, it is \( 1 - \frac{1}{6} = \frac{5}{6} \).

Key Concepts

Outcome Analysis

An essential part of understanding probability in algebra is outcome analysis. This involves determining all the possible results of a particular event. When tossing two dice, each die has 6 faces, each with a number ranging from 1 to 6.
This gives us 36 possible outcomes since each die can land on any of the 6 faces independently:

• First die (1 to 6) and second die (1 to 6)
• For every outcome on the first die, there are 6 outcomes on the second

Therefore, the combination of throwing two dice results in multiple possible outcomes, specifically 6 multiplied by 6. Using outcome analysis helps in visualizing and organizing this data effectively, laying the groundwork for calculating probabilities.

Complementary Probability

Complementary probability is a key concept when determining the likelihood of an event not happening. In our example, we are interested in calculating the probability of *not* rolling a double when tossing two dice.
To do this, first we need to find the probability of the complementary event - getting a double - which is when both dice show the same number.

• Double outcomes are limited to (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6)
• That makes 6 outcomes out of the total 36 possible outcomes

The probability of rolling a double is thus \[\frac{6}{36} = \frac{1}{6}\]The complementary probability, or the probability of not getting a double, is simply one minus the probability of getting a double: \[1 - \frac{1}{6} = \frac{5}{6}\]This approach highlights how understanding the original event allows us to easily calculate the probability of its complement.

Combinatorial Counting

Combinatorial counting is a method used to determine the total number of possible outcomes from a complex event. It often involves calculating various combinations or permutations.
When tossing two dice, each die alone offers a set of 6 outcomes. By applying the principle of combinatorial counting, we multiply these to find all the possible combinations.

• Each die with 6 outcomes means the total for two dice is 6 times 6
• This provides a total of 36 possible outcomes

This concept is crucial when analyzing probabilities because it allows for a structured approach to counting all potential scenarios. In events involving multiple stages or elements, combinatorial counting establishes a foundation that might otherwise be complex to calculate and understand.