Probability of rolling two dice

When two dice are rolled simultaneously, the possible outcomes can be represented by a sample space of 36 equally likely outcomes. Each die has 6 possible outcomes (numbers 1 to 6), and since there are two dice, the total number of possible outcomes is 6 x 6 = 36.

Identifying the favorable outcomes

To find the probability that the sum of two dice is neither 3 nor 6, we need to determine the number of favorable outcomes that satisfy this condition. Let's break it down into two parts:

Sum of 3:
There are only two ways to get a sum of 3: (1, 2) and (2, 1). These are the only favorable outcomes for a sum of 3.

Sum of 6:
Similarly, there are five ways to get a sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). These are the only favorable outcomes for a sum of 6.

Calculating the number of favorable outcomes

To find the total number of favorable outcomes, we simply add the number of favorable outcomes for a sum of 3 and a sum of 6:

Favorable outcomes = Number of favorable outcomes for sum of 3 + Number of favorable outcomes for sum of 6
Favorable outcomes = 2 + 5
Favorable outcomes = 7

Calculating the probability

The probability of an event occurring is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 7 / 36

Therefore, the probability that the sum of two dice is neither 3 nor 6 is 7/36.

Visualization:

- Total possible outcomes: 36
- Favorable outcomes for sum of 3: 2
- Favorable outcomes for sum of 6: 5
- Total favorable outcomes: 7
- Probability: 7/36