Probabilty for first 10 prime numbers pulled out of a hat

Question

Each of the first ten prime numbers is written on an individual piece of paper and put into a hat. If two are drawn at once, so that means without replacement, what is the probability that the sum of the two numbers will be prime?

Answer 1

The first 10 prime numbers are:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
 
Probability is the ratio of successes to possible outcomes.
 
There are 10 ways to select the first number, then 9 ways to select (no replacement) the second number.  So, 90 possible outcomes.
 
Then, there is an interesting property of prime numbers seen here.  Since 2 is the only even prime number and the sum of two odd numbers is always even (and thus, not prime), one of the two prime numbers being added will always be a 2.

 

For this problem, that means that if a 2 is selected, then there are 5 other numbers (3,5,11,17,29) that can be added to it, or 2 be added to, to produce a prime sum.  Thus, there are only 10 successes out of the 90 possibilities.
 
P = 10/90
P = 1/9