Question

What is the probability that the sum 11 will appear in a single throw of 2 dice?

Short answer

The probability of getting a sum of 11 in a single throw of 2 dice is \( \frac{1}{18} \).

Step-by-step solution

Identifying the range of values in dice

We know that dice have 6 faces, each with a number between 1 and 6. When we throw 2 dice, the minimum value we can get on each die is 1, and the maximum is 6.

Identifying the total number of possible outcomes

Since each die has 6 faces and we are throwing 2 dice, the total possible outcomes would be the product of 6 with itself, which is 6 x 6 = 36 possible outcomes.

Determining the outcomes that lead to a sum of 11

Now, we need to find the combinations of dice values that would give us a sum of 11. The possible combinations are: - Die 1: 5 and Die 2: 6 (5 + 6 = 11) - Die 1: 6 and Die 2: 5 (6 + 5 = 11) It is important to note that even though both combinations give us the same sum, they are considered as separate outcomes since the values on each die are different.

Calculating the probability of getting a sum of 11

Now that we have found the number of successful outcomes (2) and the total number of possible outcomes (36), we can calculate the probability as follows: Probability of getting a sum of 11 = (Number of successful outcomes) / (Total number of outcomes) = \( \frac{2}{36} \)

Simplifying the probability

The fraction \( \frac{2}{36} \) can be simplified by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2: \( \frac{2}{36} = \frac{2 \div 2}{36 \div 2} = \frac{1}{18} \) Therefore, the probability of getting a sum of 11 in a single throw of 2 dice is \( \frac{1}{18} \).

Key concepts

Probability Calculation

Understanding probability calculation is akin to grasping the odds of something occurring. In the context of the dice problem, probability essentially measures how likely it is to roll a sum of 11 with two dice. To calculate this probability, one must know the number of favorable outcomes (in this case, outcomes that sum up to 11) and the total number of possible outcomes when two dice are rolled.

Imagine each die as an independent event with 6 possible outcomes, one for each face of the die. The probability of rolling sum 11 is then computed by dividing the number of ways to achieve this sum (the favorable outcomes) by the total number of combinations when rolling two dice. As we established in the solution, only 2 combinations (5 and 6 or 6 and 5) result in a sum of 11. With 36 possible combinations (6 outcomes of the first die multiplied by 6 outcomes of the second), the probability is the ratio of the two counts.

Dice Outcomes

A dice game is a perfect example of a simple, probabilistic system. Each die has 6 faces, numbered 1 through 6. When rolling two dice, the outcomes range from 2 (1+1) to 12 (6+6). However, not all sums are equally likely. Some sums, like 7, have more possible dice combinations (six, to be exact), while others, like 11, have fewer.

To visualize dice outcomes, one can create a grid or chart with one die's outcomes on one axis and the second die's outcomes on the other axis, forming a 6x6 matrix of possibilities. By examining such a grid, students can easily identify the combinations that yield particular sums and better understand the distribution of outcomes, which is essential for solving probability problems involving dice rolls.

Simplifying Fractions

Fractions represent a part of a whole, and simplifying them is the process of reducing them to their simplest form. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator (the top number) and the denominator (the bottom number) and divide both by this number. This is useful not only in probability calculations but also in all areas of mathematics where fractions appear.

In our dice problem, the unsimplified probability of rolling a sum of 11 is written as \( \frac{2}{36} \), and the GCD of 2 and 36 is 2. By dividing both parts of the fraction by the GCD, we reduce the fraction to its simplest form, \( \frac{1}{18} \). Simplifying fractions helps in comparing probabilities and makes understanding and communicating mathematical results easier and clearer.