Understanding probability involves identifying all possible outcomes and the specific outcomes that satisfy the condition we are interested in. In this question, we are rolling two standard six-sided dice and want to find the probability that the sum of the numbers shown on the two dice is 7.

Calculating Probability for Two Dice Roll

When two dice are thrown together, the outcome is a pair of numbers, one from each die. Let's list the steps to find the probability:

1. Determine the total number of possible outcomes.
2. Determine the number of favorable outcomes (where the sum is 7).
3. Calculate the probability using the formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes).

Total Possible Outcomes When Rolling Two Dice

Each die has 6 possible outcomes (1, 2, 3, 4, 5, 6). When rolling two dice, the total number of possible combinations is the product of the outcomes for each die.

Total outcomes = Outcomes on Die 1 $\times$ Outcomes on Die 2

Total outcomes = $6 \times 6 = 36$

These 36 outcomes form the sample space. We can represent them as pairs (Die 1 result, Die 2 result).

Favorable Outcomes: Getting a Sum of 7

We need to find the pairs of outcomes from the 36 total possibilities where the sum of the two numbers is exactly 7. Let's list them:

• If the first die is 1, the second must be 6 (1 + 6 = 7). Outcome: (1, 6)
• If the first die is 2, the second must be 5 (2 + 5 = 7). Outcome: (2, 5)
• If the first die is 3, the second must be 4 (3 + 4 = 7). Outcome: (3, 4)
• If the first die is 4, the second must be 3 (4 + 3 = 7). Outcome: (4, 3)
• If the first die is 5, the second must be 2 (5 + 2 = 7). Outcome: (5, 2)
• If the first die is 6, the second must be 1 (6 + 1 = 7). Outcome: (6, 1)

Counting these favorable outcomes, we find there are 6 ways to get a sum of 7 when rolling two dice.

Here is a table illustrating all possible sums when rolling two dice, highlighting the sums of 7:

Calculating the Probability

Now we use the probability formula:

Probability (Sum of 7) = (Number of outcomes with sum 7) / (Total number of possible outcomes)

Probability (Sum of 7) = 6 / 36

This fraction can be simplified, but the options are given in the form of x/36. So, the probability of getting a total of 7 on two dice thrown together is 6/36.

Probability Solution Summary

• Total possible outcomes: 36 (pairs from (1,1) to (6,6))
• Favorable outcomes (sum is 7): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - there are 6 such outcomes.
• Probability = Favorable Outcomes / Total Outcomes = 6 / 36.

This matches option 1.

Revision Table: Probability on Two Dice

Additional Information on Probability Basics

Probability is a measure of the likelihood of an event occurring. It is calculated as:

$$P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Key points about probability:

• The value of probability always ranges from 0 to 1, inclusive. 0 means the event is impossible, and 1 means the event is certain.
• The sum of probabilities of all possible mutually exclusive outcomes in a sample space is always 1.
• Understanding sample space is crucial. It is the set of all possible outcomes of a random experiment. For two dice, the sample space size is 36.
• Events are subsets of the sample space. In this question, the event is "getting a sum of 7".