Problem 44 - College Algebra with Modeling and Visualization

Question

Two dice are rolled. Find the probability that the dice show a sum other than 7 or 11

Step-by-Step Solution

Answer

The probability is \(\frac{7}{9}\).

Step 1: Determine Total Outcomes

When two dice are rolled, each die has 6 faces. Therefore, the total number of possible outcomes is 6 * 6 = 36.

Step 2: List Outcomes with Sum 7 or 11

List the outcomes where the sum of the dice is either 7 or 11. - Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - making 6 outcomes. - Sum of 11: (5,6), (6,5) - making 2 outcomes. Thus, there are 6 + 2 = 8 outcomes where the sum is 7 or 11.

Step 3: Calculate Favorable Outcomes

To find the probability of a sum other than 7 or 11, we must count the outcomes that do not result in a sum of 7 or 11. The number of favorable outcomes is therefore 36 total outcomes - 8 outcomes with sum 7 or 11 = 28 outcomes.

Step 4: Calculate the Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.\[P(\text{sum other than 7 or 11}) = \frac{28}{36} = \frac{7}{9}\]

Step 5: Simplify the Probability

Simplify the fraction \(\frac{28}{36}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4.\[\frac{28}{36} = \frac{7}{9}\]Thus, the probability that the dice show a sum other than 7 or 11 is \(\frac{7}{9}\).

Key Concepts

Dice Outcomes

When working with dice, it's essential to understand the concept of outcomes. A standard die has six faces numbered from 1 to 6.
When you roll two dice, each die operates independently, creating various combinations.
This independence allows each die to have its full range of outcomes no matter what the other die shows.To determine all the possible outcomes when rolling two dice, we must consider all combinations of the two dice.

• For the first die, there are 6 possible outcomes.
• For the second die, also 6 possible outcomes exist.

Therefore, when calculating possible outcomes for both dice together, you multiply the number of outcomes for each die:
\[6 \times 6 = 36\]
This equation shows that rolling two dice results in 36 possible combinations or outcomes.

Probability Calculation

When calculating probability, we evaluate how likely an event is to occur compared to all other possibilities. If you're rolling two dice and want to find the probability of not rolling a sum of 7 or 11, the process involves several steps.First, list all combinations that result in a sum of 7 or 11.

• Sum of 7 includes these pairs: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
• Sum of 11 includes these pairs: (5,6) and (6,5).

Adding these outcomes gives us 8 combinations.
Thus, to find how many outcomes result in a sum other than 7 or 11, subtract these 8 outcomes from the 36 total outcomes:
\[36 - 8 = 28\]With these facts, calculate the probability by comparing these 28 favorable outcomes to the 36 total possibilities:
\[P(\text{sum other than 7 or 11}) = \frac{28}{36}\]

Simplifying Fractions

Simplifying fractions is a crucial exercise in making probabilities clearer and more understandable. Mathematical probability often starts as a fraction. In many instances, like in this example, the initial calculation results in
\(\frac{28}{36}\).

To simplify, you'll need to find the greatest common divisor (GCD) of both the numerator and denominator. The GCD of 28 and 36 is 4.
To simplify, divide both by this number:\[\frac{28 \div 4}{36 \div 4} = \frac{7}{9}\]So, the simplified probability of rolling a sum other than 7 or 11 with two dice becomes \(\frac{7}{9}\). Simplified fractions provide a clearer and more concise representation of probability, making it easier to compare with other ratios.