Here’s the answer from Monday’s Brain Twister!
A math teacher assigns a distinct prime number, starting with the lowest, for each student in her class. If she chooses two students at random, the probability the sum of their numbers is not even is less than 1/10. What is the fewest possible number of students in her class?
A good way to approach this question is by back solving. That is, take one of the answer choices and “put it back” into the question. For instance, let’s assume the answer is (B) 20, which is a nice easy number to work with. If there are 20 students in the classroom, one of them has to be assigned the number 2. This is important because ‘2’, the only even prime, is the only way to obtain an odd number when summing two primes (Odd + Even = Odd).
The chances of choosing that student given 20 total students, 19 of whom have odd prime numbers, can be described as the following:
# of desired outcomes
# of total outcomes
The one student with an even number can be paired with 19 other students, giving us 19 in the numerator.
The total number of pairings can be found by using combinations, not permutations, since we don’t want to count the same student twice (Student 1 and Student 2 in a group is the same as Student 2 and Student 1 in a group).
20!/18!2! = 190 = # of total outcomes
19/190 = 1/10
We know that the odds have to be less than 1/10. By adding one more student the odds the probability would have to dip ever so slightly below 1/10, since the more students you have in the class the less likely you’ll select the one and only student who has the even prime number, 2.
Therefore, by adding that one extra student we will have a probability that is slightly less than 1/10, meaning that 21 is the fewest number of students we can have in the class. Answer: (C).
I should mention one more thing: the question doesn’t specify whether the primes after the lowest prime, the number 2, are in specific order. But since we are only dealing with the odd and even properties of prime numbers, and every prime number after ‘2’ is odd, it doesn’t matter which primes those are.