If two fair dice are rolled, what is the probability that a total showing is either even or less than eight?
Thanks for your help
To find the probability of getting a total that is either even or less than eight when rolling two fair dice, we need to first determine the number of favorable outcomes (outcomes that meet the condition) and the total number of possible outcomes.
Step 1: Counting the favorable outcomes
To find the number of favorable outcomes, we need to count all the pairs of numbers on the dice that satisfy either condition. Let's break it down.
1. Pairs that have an even total:
- (1, 1) → total = 2
- (1, 3) → total = 4
- (1, 5) → total = 6
- (2, 2) → total = 4
- (2, 4) → total = 6
- (2, 6) → total = 8
- (3, 1) → total = 4
- (3, 3) → total = 6
- (3, 5) → total = 8
- (4, 2) → total = 6
- (4, 4) → total = 8
- (4, 6) → total = 10
- (5, 1) → total = 6
- (5, 3) → total = 8
- (5, 5) → total = 10
- (6, 2) → total = 8
- (6, 4) → total = 10
- (6, 6) → total = 12
In total, there are 18 pairs that give an even total.
2. Pairs that have a total less than 8 but are not even:
- (1, 2) → total = 3
- (1, 4) → total = 5
- (2, 1) → total = 3
- (2, 3) → total = 5
- (3, 2) → total = 5
- (4, 1) → total = 5
- (4, 3) → total = 7
- (5, 2) → total = 7
- (5, 4) → total = 9
- (6, 1) → total = 7
- (6, 3) → total = 9
In total, there are 11 pairs that have a total less than 8 but are not even.
Step 2: Counting the total number of outcomes
To find the total number of outcomes, consider that each die has six possible outcomes (numbers 1 to 6). Since we are rolling two dice, the total number of outcomes is calculated by multiplying the number of outcomes on each die. So, the total number of outcomes is 6 x 6 = 36.
Step 3: Calculating the probability
The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. So, the probability is (18 + 11) / 36.
Final answer:
The probability that the total showing on two fair dice is either even or less than eight is (18 + 11) / 36 = 29 / 36.