Roszena rolls two standard six-sided dice. If the sum of the numbers on the top faces is not 6, what is the probability that the sum is 7? Express your answer as a rational number in lowest terms.
The number of combinations that does NOT sum up to 6 is 31. (1-1,1-2,1-3,1-4,1-6;2-1,2-2,2-3,2-5,2-6; 3-1,3-2,3-4,3-5,3-6;4-1,4-3,4-4,4-5,4-6;
5-2,5-3,5-4,5-5,5-6;6-1,6-2,6-3,6-4,6-5, and 6-6)
Six of those combinations add up to 7. Therefore the probability is 6/31.
To find the probability of rolling a sum of 7 when the sum is not 6, we need to determine the sample space and the favorable outcomes.
The sample space consists of all the possible outcomes when rolling two standard six-sided dice. Since each die has 6 sides, the total number of outcomes is 6 * 6 = 36.
Next, we need to find the number of favorable outcomes where the sum is 7 and the sum is not 6. To do this, we can list all the possible outcomes:
(1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1)
Out of these, (1, 6) and (6, 1) have a sum of 7, but a sum of 6 as well. Therefore, there are 4 favorable outcomes where the sum is 7 but not 6.
So, the probability is given by the number of favorable outcomes divided by the sample space:
Probability = Number of favorable outcomes / Sample space
Probability = 4 / 36
To simplify this fraction, we can find the greatest common divisor (GCD) of 4 and 36, which is 4. Dividing both the numerator and denominator by the GCD, we get:
Probability = 1 / 9
Therefore, the probability of rolling a sum of 7, given that the sum is not 6, is 1/9.