Even Prime Sum

List the primes between 1 and 20 and note that only one of them is even.

The sum is odd exactly when one prime is 2 and the other is odd.

1. 1. List primes

   Primes from 1 to 20: 2,3,5,7,11,13,17,19 — 8 primes, of which only 2 is even.

2. 2. Total pairs

   Total ways to choose 2 distinct primes: $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$.

3. 3. Odd-sum pairs

   Sum is odd only if one prime is 2: 7 such pairs. So even-sum pairs are $28-7=21$.

4. 4. Probability

   Probability of even sum is $21/28=3/4$.

   $$P=\frac{21}{28}=\frac{3}{4}$$

Answer: 3/4