Sam rolls a standard die, and then Ruth rolls the same die. The probability that Ruth's number is higher than Sam's can be written as a/b where a and b have no common factors greater than 1. Find a.

There are six numbers, 36 boxes below

* 1 2 3 4 5 6 Sam
1
2 *
3 * *
4 * * *
5 * * * *
6 * * * * *
Ruth
(5 + 4 + 3 + 2 +1)/36 = 15/36 = 5/12

5. The probability that both Sam’s

and Ruth’s roll produce the same result
is 1/6. The likelihood that Ruth has the
higher number is the same as the likelihood that Sam has the higher number.
Thus, we divide the remaining 5/6 in
half to get 5/12; the numerator is 5.

To find the probability that Ruth's number is higher than Sam's, we need to find the number of favorable outcomes and the total number of possible outcomes.

Since Sam rolls the die first, he can get any number from 1 to 6 with equal probability. For each of these numbers, there are (6-1) = 5 numbers that Ruth can roll to have a higher number.

Therefore, the total number of favorable outcomes is 6 * 5 = 30.

The total number of possible outcomes is 6 * 6 = 36, since both Sam and Ruth can get any number from 1 to 6.

So, the probability that Ruth's number is higher than Sam's is 30/36.

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 6. Dividing both 30 and 36 by 6 gives 5/6.

Therefore, a = 5.

To find the probability that Ruth's number is higher than Sam's, we need to consider all the possible outcomes when rolling a standard die.

Step 1: Determine the total number of outcomes
A standard die has 6 sides, numbered from 1 to 6. Therefore, the total number of outcomes when rolling a die is 6.

Step 2: Determine the number of favorable outcomes
For Ruth's number to be higher than Sam's, we need to count the number of outcomes where Ruth's number is greater than Sam's number. Let's list them:

- Sam rolls 1 (S1) and Ruth rolls 2, 3, 4, 5, or 6 (R2, R3, R4, R5, R6).
- Sam rolls 2 (S2) and Ruth rolls 3, 4, 5, or 6 (R3, R4, R5, R6).
- Sam rolls 3 (S3) and Ruth rolls 4, 5, or 6 (R4, R5, R6).
- Sam rolls 4 (S4) and Ruth rolls 5 or 6 (R5, R6).
- Sam rolls 5 (S5) and Ruth rolls 6 (R6).

Counting all these outcomes, we find a total of 15 favorable outcomes.

Step 3: Calculate the probability
The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes.
P(Ruth's number > Sam's number) = favorable outcomes / total outcomes = 15/6

Now, let's simplify the fraction 15/6:

- First, we can divide both the numerator and the denominator by their greatest common divisor, which is 3.
15/6 = (15 ÷ 3) / (6 ÷ 3) = 5/2

Therefore, a = 5.

So, the probability that Ruth's number is higher than Sam's is 5/6.