If two six-sided dice are rolled, what is the probability that the total number of spots showing is 7?

Verified step by step guidance

1. Understand the problem: You are rolling two six-sided dice, and you need to calculate the probability that the sum of the numbers on the two dice equals 7.
2. Determine the total number of possible outcomes: Each die has 6 faces, so the total number of outcomes when rolling two dice is 6 × 6 = 36.
3. Identify the favorable outcomes: List all the pairs of numbers on the two dice that add up to 7. These pairs are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are 6 favorable outcomes.
4. Calculate the probability: The probability of an event is given by the formula: $\frac{\mathrm{Number} \mathrm{of} \mathrm{favorable} \mathrm{outcomes}}{\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{outcomes}}$. Substitute the values: $\frac{6}{36}$.
5. Simplify the fraction to express the probability in its simplest form.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of rolling dice, it quantifies the chance of obtaining a specific outcome, such as the sum of the numbers showing on the dice. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For rolling two six-sided dice, the sample space consists of all pairs of numbers that can appear, ranging from (1,1) to (6,6). This results in a total of 36 possible outcomes, which is crucial for calculating probabilities related to specific sums, such as 7.

Favorable Outcomes

Favorable outcomes refer to the specific results that satisfy the condition of the probability question. In this case, to find the probability of rolling a total of 7 with two dice, we identify all combinations that yield this sum, such as (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are six favorable outcomes, which are essential for calculating the probability.