If a number is selected at random from each of the sets p = {1, 2, 3} and Q = {2, 3, 5}, find the probability that the sum of the numbers is prime

Analyze the question: One number is picked from P = {1, 2, 3} and one from Q = {2, 3, 5}. We need the probability that their sum is a prime number.

Concept: Probability = (number of favourable outcomes) ÷ (total number of outcomes). The total is 3 × 3 = 9 equally likely pairs.

Solution: List all the sums.

| P \ Q | 2 | 3 | 5 | |---|---|---|---| | 1 | 3 | 4 | 6 | | 2 | 4 | 5 | 7 | | 3 | 5 | 6 | 8 |

Now pick out the prime sums (2, 3, 5, 7, 11, ...):

• 1+2 = 3 (prime)
• 1+3 = 4 (not prime)
• 1+5 = 6 (not prime)
• 2+2 = 4 (not prime)
• 2+3 = 5 (prime)
• 2+5 = 7 (prime)
• 3+2 = 5 (prime)
• 3+3 = 6 (not prime)
• 3+5 = 8 (not prime)

Prime sums: 3, 5, 7, 5, which gives 4 favourable outcomes.

P(sum is prime) = 4/9

Tip: For "sum of two selections" questions, always draw the sum table. It stops you from missing outcomes. Also remember that 1 is not prime, and 2 is the only even prime.