Question

Two fair dice are tossed. a. What is the probability that the sum of the number of dots shown on the upper faces is equal to 7 ? To \(11 ?\) b. What is the probability that you roll "doubles" that is, both dice have the same number on the upper face? c. What is the probability that both dice show an odd number?

Short answer

Answer: a) The probability of the sum being 7 or 11 is \(\frac{2}{9}\). b) The probability of rolling doubles is \(\frac{1}{6}\). c) The probability of both dice showing odd numbers is \(\frac{1}{4}\).

Step-by-step solution

Find Total Number of Possible Outcomes

When two fair dice are rolled, each die can show any number from 1 to 6. Therefore, there are a total of \(6 \times 6 = 36\) possible outcomes.

Calculate Probability of the Sum Being 7 or 11

To determine the probability of the sum of the number of dots being equal to 7 or 11, we first need to find the number of favorable outcomes. For the sum to be 7, we have the following outcomes: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). For the sum to be 11, we have: (5, 6), and (6, 5). This gives us a total of 8 favorable outcomes. Now, we can calculate the probability: \(\frac{8}{36} = \frac{2}{9}\).

Calculate Probability of Rolling Doubles

For rolling doubles, both dice have the same number on the upper face. We have the following possible outcomes: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6). So, there are 6 favorable outcomes. Therefore, the probability of rolling doubles is: \(\frac{6}{36} = \frac{1}{6}\).

Calculate Probability of Both Dice Showing Odd Numbers

For both dice to show odd numbers, we have the following possible outcomes: (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), and (5, 5). So, there are 9 favorable outcomes. Now, we can calculate the probability: \(\frac{9}{36} = \frac{1}{4}\). To summarize: a. The probability of the sum being 7 or 11 is \(\frac{2}{9}\). b. The probability of rolling doubles is \(\frac{1}{6}\). c. The probability of both dice showing odd numbers is \(\frac{1}{4}\).

Key concepts

Dice outcomes

When rolling two dice, each die has an independent chance of landing on any number between 1 and 6. This means, for each roll of the die, you have six possible outcomes. When two dice are rolled together, we need to consider the combined outcomes of both dice. Each number on one die can pair with any of the six numbers on the other die.
To find the total number of outcomes when rolling two dice, we multiply the number of possibilities for one die by the number of possibilities for the other die. This gives us a formula:

• Total outcomes = 6 outcomes (Die 1) × 6 outcomes (Die 2) = 36 outcomes

Visualizing this, you could draw a grid with one die's outcomes along the rows and the other die's outcomes along the columns, resulting in all possible pairs or combinations. This grid helps to understand why there are 36 outcomes.

Probability calculations

Probability is a measurement of how likely an event is to happen. When calculating probability, we use the formula:

• Probability = \(\frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}}\)

Here, for dice, the total possible outcomes are 36, as each die has 6 sides. If you want to find how probable it is to roll a certain result or combination, it's key to understand how many of those combinations are 'favorable', fitting your desired condition. For example, to calculate the probability of getting a certain sum or doubles, knowing both the total and favorable outcomes helps to determine the probability.

Favorable outcomes

Favorable outcomes are specific results that align with the condition or event we are interested in. For instance, if you are keen on rolling a sum of 7, favorable outcomes are the combinations of dice that sum up to 7, such as (1, 6), (2, 5), etc. For two dice, finding all these combinations systematically involves checking each pair that meets this condition, which might include some intuitive or remembered pairs.
In the case of rolling a sum of 11, the pairs that meet this are (5, 6) and (6, 5). The more familiar you become with possible dice combinations, the easier it becomes to identify favorable outcomes without enumerating all possibilities each time.

Sum of dice

The sum of dice refers to the total number of dots shown on the upper faces of two rolled dice. With numbers ranging from 2 (1+1) to 12 (6+6), achieving different sums depends on the specific combinations of numbers that add up to the target sum.
Let's explore how many ways we can roll specific sums:

• A sum of 7 can be rolled with (1, 6), (2, 5), (3, 4), and more.
• A sum of 11 can only be obtained with (5, 6) or (6, 5).

Understanding all possible dice outcomes helps us see that some sums, like 7, have more combinations and therefore are more probable than others, like 2 or 12.

Rolling doubles

Rolling doubles occurs when both dice show the same number. For example, rolling (1, 1), (2, 2), through to (6, 6) counts as doubles. These combinations are symmetric, and each pair counts as a favorable outcome within the possible 36 outcomes of two dice.

• The favorable outcomes for doubles are:
  • (1, 1),
  • (2, 2),
  • (3, 3),
  • (4, 4),
  • (5, 5),
  • (6, 6)

Since there are 6 favorable outcomes where doubles appear, we calculate the probability simply as \(\frac{6}{36} = \frac{1}{6}\). Recognizing these pairs helps in quickly determining the likelihood of rolling doubles during a game.