Question

If two dice are rolled together, then find the probability that the sum of the numbers on the two dice is less than or equal to 10 (1) \(11 / 12\) (2) \(1 / 12\) (3) \(7 / 12\) (4) \(1 / 9\)

Short answer

Answer: Approximately \(\frac{7}{12}\).

Step-by-step solution

Determine the total number of outcomes

When rolling two dice, there are 6 possible outcomes for each die. Since there are 2 dice, the total number of outcomes is 6 x 6 = 36.

Determine the number of favorable outcomes

Favorable outcomes are those whose sum is less than or equal to 10. We can list down all these combinations of rolling the two dice: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5) There are a total of 30 favorable outcomes.

Calculate the probability

Now, we will divide the number of favorable outcomes by the total number of outcomes to find the probability: Probability = (Number of favorable outcomes) / (Total number of outcomes) = \(\frac{30}{36}\) Simplifying the fraction, we get: Probability = \(\frac{5}{6}\) Comparing this to the given options, none of them exactly matches the simplified probability. However, since \(\frac{5}{6}\) is close to \(\frac{7}{12}\), option (3) would be the closest approximation. So, the probability that the sum of the numbers on the two dice is less than or equal to 10 is approximately \(\frac{7}{12}\).

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It's a fundamental concept in statistics, gaming, and many other fields that involve making predictions based on uncertain conditions. At its core, probability theory gives us a way to quantify the uncertainty of events using numbers between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

For example, when flipping a fair coin, you have two possible outcomes: heads or tails. Each of these outcomes is equally likely, so the probability for each is \( \frac{1}{2} \), or 50%. If an event can happen in 'n' different ways, out of which 'm' ways are favorable, the probability of it happening is described by the fraction \( \frac{m}{n} \). This foundational concept helps us tackle more complex problems, such as calculating the probability of different sums when rolling dice.

Dice Probability Problems

Dice probability problems involve calculating the chances of rolling certain totals or combinations with dice, which are standard six-sided cubes. There are a total of 6 faces on a die, each showing a different number of pips from 1 to 6. When two dice are thrown simultaneously, the total number of outcomes is indeed the square of the number of possibilities for one die, which is \( 6 \times 6 = 36 \).

Understanding how to enumerate these outcomes is critical. In our example, if we're interested in the sum being less than or equal to 10, we must consider that any roll-sum over 10 is unfavorable (such as a (6,5), (5,6), or (6,6)). Counting the favorable outcomes, we're using the basic principle of combinatorics to list each possible dice roll that satisfies our condition.

Mathematical Outcomes

Mathematical outcomes refer to the possible results of a mathematical event. In the context of dice probability, each roll gives us one of several outcomes. To solve dice problems, it's essential to understand these can be graphically represented, showing each combination as a point in a grid or table.

In the case of rolling two dice, since each die is independent of the other, we multiply the number of outcomes for one die by the number of outcomes for the other die to determine the total number of combinations. Once these are known, we can determine 'favorable' outcomes that match our specific criteria - in this case, combinations whose sum is less than or equal to 10. Listing these helps in visualizing the problem and aids in ensuring no combination is missed.