Question

If two 6-sided dice are thrown, what is the probability that the sum of the faces is \(7 ?\)

Short answer

The probability is \( \frac{1}{6} \).

Step-by-step solution

Calculate Total Possible Outcomes

Each die has 6 faces, so if two dice are thrown, the total number of possible outcomes is calculated by multiplying the number of faces on each die together: \[ 6 \times 6 = 36 \]There are 36 possible outcomes when two 6-sided dice are thrown.

Identify Outcomes That Sum to 7

List all the pairs of dice faces that sum to 7: \[ (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) \]There are 6 pairs that result in the sum of 7.

Calculate the Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For this problem, the probability is: \[ \frac{6}{36} = \frac{1}{6} \]The probability that the sum of the faces is 7 when two 6-sided dice are thrown is \( \frac{1}{6} \).

Key concepts

probability calculations

Probability calculations help you determine how likely an event is to happen. For any event, it's expressed as a fraction, where the numerator is the number of favorable outcomes, and the denominator is the total number of possible outcomes. In our example, we sought the probability of getting a sum of 7 when rolling two 6-sided dice.
First, calculate the total possible outcomes. Each die has 6 faces, so with two dice, you have 36 potential combinations: \[6 \times 6 = 36\].
Then, identify the favorable outcomes that form the sum of 7: \[(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\].
Six combinations give us 7. So you divide the number of favorable outcomes by the total outcomes: \[\frac{6}{36} = \frac{1}{6}\].
It's essential to approach probability calculations methodically to ensure accuracy.

GMAT math preparation

Preparing for the GMAT involves mastering various mathematical concepts, including probability. GMAT questions often focus on understanding and applying basic principles of probability.
When preparing, practice is key. Work on a variety of problems to get comfortable with different scenarios and calculations. Pay special attention to:

• Basic probability principles
• Combinations and permutations
• Real-world probability applications

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In our dice example, you use these fundamental concepts to find a specific outcome’s likelihood, which is common in GMAT questions. Deepen your understanding through repetition and by breaking problems into smaller, manageable steps.

probability of dice outcomes

Dice are a classic tool for studying probability. A standard 6-sided die has equal chances for each face to land facing up, making it a great example for probability lessons.
For two 6-sided dice, calculating the outcomes involves understanding how independent events combine. The total number of outcomes is found by multiplying the possibilities for each die: \[6 \times 6 = 36\]
Next, you identify specific events of interest. To find the probability of summing to 7, you count the matching combinations: \[(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\].
You then express this as a fraction of the total outcomes: \[\frac{6}{36} = \frac{1}{6}\]
This classic problem reinforces the method of counting favorable outcomes and dividing by the total number to find probabilities.