Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction.
There are $\binom{20}{2}$ pairs of distinct integers between 1 and 20, and there are $\binom{8}{2}$ pairs of distinct prime numbers between 1 and 20.  Therefore, the probability that both members of a randomly chosen pair are prime is $\dfrac{\binom{8}{2}}{\binom{20}{2}}=\dfrac{8(7)/2}{20(19)/2}=\boxed{\dfrac{14}{95}}$.
\dfrac{14}{95}
Level 3
Counting & Probability
Problem: Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability that both chosen integers are prime, we first identify the prime numbers between 1 and 20. The prime numbers in this range are: 2, 3, 5, 7, 11, 13, 17, and 19. This gives us a total of 8 prime numbers.

Next, we calculate the total number of ways to choose 2 different integers from the set of integers 1 through 20. The total number of integers in this range is 20. The number of ways to choose 2 integers from 20 is given by the combination formula C(n, k), where n is the total number of items to choose from, and k is the number of items to choose. Thus, we calculate:

C(20, 2) = 20! / (2!(20-2)!) = 20! / (2! * 18!) = (20 * 19) / (2 * 1) = 190.

Now, we need to determine the number of ways to choose 2 prime numbers from the 8 identified. Using the combination formula again, we find:

C(8, 2) = 8! / (2!(8-2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28.

The probability that both chosen integers are prime is then the ratio of the number of favorable outcomes (choosing 2 primes) to the total outcomes (choosing any 2 integers):

Probability = C(8, 2) / C(20, 2) = 28 / 190.

To simplify this fraction, we find the greatest common divisor (GCD) of 28 and 190. The GCD is 2. Dividing both the numerator and the denominator by 2 gives:

28 / 2 = 14 and 190 / 2 = 95.

Thus, the simplified probability that both numbers chosen are prime is:

14 / 95.<|END_OF_LATENT|> The final answer is $\dfrac{14}{95}$.