To calculate the probability of rolling a sum of 8 with two number cubes (standard six-sided dice), we need to determine two things:
1. The total number of possible outcomes when rolling two dice.
2. The number of successful outcomes where the sum of the two dice equals 8.
Let's solve these step by step:
**Step 1: Calculating Total Possible Outcomes**
Each die has 6 faces, so when rolling one die, there are 6 possible outcomes. When rolling two dice, each die operates independently of the other, so for each outcome of the first die, there are 6 possible outcomes of the second die. Therefore, the total possible outcomes are calculated by multiplying the number of outcomes for one die by the number of outcomes for the second die.
Total outcomes = 6 (from the first die) * 6 (from the second die) = 36
**Step 2: Finding the Number of Successful Outcomes**
A successful outcome is one where the sum of the two dice equals 8. So we need to find all the possible pairs of numbers on the two dice that would sum up to 8. These pairs are:
- Die 1 rolls a 2 and Die 2 rolls a 6 (2 + 6 = 8).
- Die 1 rolls a 3 and Die 2 rolls a 5 (3 + 5 = 8).
- Die 1 rolls a 4 and Die 2 rolls a 4 (4 + 4 = 8).
- Die 1 rolls a 5 and Die 2 rolls a 3 (5 + 3 = 8).
- Die 1 rolls a 6 and Die 2 rolls a 2 (6 + 2 = 8).
Note that since the dice are distinct, the pair (2, 6) is different from the pair (6, 2). Same goes for the pairs (3, 5) and (5, 3).
Therefore, the number of successful outcomes = 5 (since there are 5 different pairs that sum to 8).
**Step 3: Calculating the Probability**
The probability of an event is calculated as the number of successful outcomes divided by the total possible outcomes.
Probability = Number of successful outcomes / Total possible outcomes
Probability = 5 / 36
Thus the probability of rolling two number cubes and getting a sum of 8 is 5/36.
