Question

(a) What is the probability of rolling a pair of dice and obtaining a total score of 9 or more? (b) What is the probability of rolling a pair of dice and obtaining a total score of \(7 ?\)

Short answer

Probability of 9 or more: \(\frac{5}{18}\); Probability of 7: \(\frac{1}{6}\).

Step-by-step solution

Understand the Problem

We need to calculate the probability of two different outcomes when rolling a pair of dice: obtaining a total score of 9 or more, and obtaining a total score of 7. We must consider all possible combinations of the dice roles that meet these criteria.

Calculate Total Possible Outcomes

When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is given by multiplying the number of outcomes for each die: \(6 \times 6 = 36\).

Identify Favorable Outcomes for a Total of 9 or More

List the pairs of numbers that sum to 9 or more. These include: (3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), and (6,6). This gives us a total of 10 outcomes.

Calculate Probability for a Total of 9 or More

Use the probability formula: \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \). For a total of 9 or more: \( \frac{10}{36} = \frac{5}{18} \approx 0.2778 \).

Identify Favorable Outcomes for a Total of 7

List the pairs of numbers that sum to exactly 7. These include: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). This gives us a total of 6 outcomes.

Calculate Probability for a Total of 7

Use the probability formula for a total of 7: \( \frac{6}{36} = \frac{1}{6} \approx 0.1667 \).

Conclusion

We have calculated the probabilities for both scenarios: obtaining a total score of 9 or more, and obtaining a total score of 7.

Key concepts

Rolling Dice

Rolling dice is a fun way to explore probabilities because each die has an equal chance of landing on any of its six faces: 1, 2, 3, 4, 5, or 6. When you roll a single die, there are 6 possible outcomes. However, when you roll two dice simultaneously, the number of possible outcomes increases exponentially.

To calculate all potential results of two dice, consider the combinations formed by each die's possible outcomes. A roll of two dice can produce 36 different pairs, calculated by multiplying the 6 faces of one die by the 6 faces of the other die. This is why the total number of possible outcomes when rolling two dice is expressed as:

• \(6 \times 6 = 36\)

This fundamental step paves the way to understanding probabilities, as it sets the stage for considering both favorable outcomes and the probability formula.

Favorable Outcomes

Favorable outcomes are the specific results that meet our criteria or conditions for a probability question. When rolling two dice, we may want certain pairs of numbers. For example, in our problem, we sought outcomes totaling to 9 or more, and exactly 7.

To identify these, list each die's face values that add up to meet your condition. For example:

• For a sum of 9 or more: pairs like (3,6), (4,5), and (6,6) are included, resulting in 10 favorable outcomes.
• For a sum exactly 7: pairs like (1,6), (3,4), and (6,1) fall into this category, giving us 6 favorable outcomes.

By identifying favorable outcomes, you can move closer to solving for the probability of your event occurring.

Probability Formula

The probability of any event happening is a comparison between favorable outcomes and the total possible outcomes. This is achieved through the probability formula. Probability is expressed as:

• \( ext{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}}\)

Using this, you can determine the likelihood of an event. For a total roll of 9 or more:

• \( \frac{10}{36} = \frac{5}{18} \approx 0.2778\), meaning roughly a 27.78% chance.

For a total roll of exactly 7:

• \( \frac{6}{36} = \frac{1}{6} \approx 0.1667\), indicating about a 16.67% chance.

By applying this formula, you can find probabilities for any given condition with dice or similar scenarios.

Total Possible Outcomes

Understanding total possible outcomes is crucial when calculating probability. When rolling two dice, each die holds 6 potential outcomes. These combine to form a matrix of 36 possible outcomes when considering a pair of dice.

• Each die has faces: 1, 2, 3, 4, 5, and 6.
• Possible combinations range from (1,1) to (6,6).

This comprehensive consideration of outcomes ensures no possibilities are overlooked and provides a solid foundation for probability calculations. The total of 36 outcomes is a constant factor against which favorable outcomes are compared. Recognizing this step is key to mastering the computation of probabilities with dice and beyond.