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


greater than 5?

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 is greater than 5.
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. List all possible outcomes where the sum of the two dice is greater than 5. For example, if the first die shows 1, the second die must show at least 5 to achieve a sum greater than 5. Repeat this for all possible values of the first die (1 through 6).
4. Count the number of favorable outcomes where the sum is greater than 5. This involves systematically identifying all pairs (x, y) where x and y are the numbers on the dice, and x + y > 5.
5. Calculate the probability by dividing the number of favorable outcomes by the total number of outcomes. Use the formula: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).

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 this context, it quantifies the chance of rolling a total greater than 5 with two six-sided dice. The probability 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 36 combinations (6 sides on the first die multiplied by 6 sides on the second die). Understanding the sample space is crucial for calculating probabilities accurately.

Favorable Outcomes

Favorable outcomes refer to the specific results that satisfy the condition of the probability question. In this case, we need to count how many combinations of the two dice yield a total greater than 5. Identifying these outcomes is essential for determining the probability of the event.