Question

Two dice are rolled together, what is the probability that the total score on the two dice is greater than \(10=\) (1) \(\frac{5}{6}\) (2) \(\frac{1}{4}\) (3) \(\frac{1}{6}\) (4) \(\frac{1}{12}\)

Short answer

Answer: \(\frac{1}{12}\)

Step-by-step solution

Identify all possible outcomes

When two dice are rolled, there are a total of 36 possible outcomes as each dice has 6 sides and can produce a number between 1 to 6. To list all possible outcomes, we can represent each dice roll as an ordered pair (x, y) where x and y are integers from 1 to 6.

Count the favorable outcomes

Now, we have to find the number of outcomes where the sum of the numbers on the rolled dice is greater than 10. These include (5, 6), (6, 5), (6, 6). So, there are 3 favorable outcomes.

Calculate the probability

Finally, we will calculate the probability of getting a sum greater than 10. The probability is the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability = \(\frac{3}{36} = \frac{1}{12}\). Hence, the correct answer is (4) \(\frac{1}{12}\).

Key concepts

Dice Roll

Rolling dice is a classic example of a random experiment in probability. When we roll two dice, each die has 6 faces numbered from 1 to 6.
The result of a roll is a combination of the numbers appearing on both dice. We can think of each roll as an ordered pair, where the first number is the result from the first die and the second number from the second die.
For example, if the first die shows a 3 and the second a 5, the pair is (3, 5). Understanding this combination is crucial, as it helps us determine all possible results from the dice roll and aids in calculating probabilities.

Favorable Outcomes

In probability, determining favorable outcomes is essential to solve any problem. Here, we're interested in finding the outcomes where the sum of two dice is greater than 10.
To find these, we analyze combinations such as (5, 6), (6, 5), and (6, 6). Only these three combinations, out of the possible 36, result in a sum greater than 10.
Identifying favorable outcomes means isolating the specific results that satisfy the condition of our probability problem. This concept helps us focus only on what we need to find the solution.

Probability Calculation

Once favorable outcomes are identified, we calculate the probability to see how likely they are to occur. Probability is found using a simple formula:

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

In our example of rolling two dice, there are 36 possible outcomes and 3 favorable ones.
Substituting these values, we calculate the probability as \( \frac{3}{36} \), which simplifies to \( \frac{1}{12} \).
This shows that there's a one in twelve chance of rolling a sum greater than 10 with two dice.

Outcomes in Probability

Understanding the outcomes in probability gives us the foundation to calculate probabilities. An outcome is simply a result of a trial, such as rolling a die. When both dice are rolled, each distinct pair (x, y) from (1, 1) to (6, 6) represents an outcome.
These outcomes form the set of all possible results, known in probability terms as the sample space.
In this exercise, by comprehending that there are 36 outcomes for two dice, we can efficiently explore which results fit our criteria of having a sum greater than 10. Knowing how to list and identify outcomes lets us apply probability rules accurately and understand the likelihood of events within any given set.