Problem Solving

Set S consists of all prime integers less than 10. If two numbers are chosen form set S at random, what is the probability that the product of these numbers will be greater than the product of the numbers which were not chosen?

A. 1/3
B. 2/3
C. 1/2
D. 7/10
E. 4/5

HI Guru/ Experts

S = {2. 3, 5, 7}

2*3=6
2*5=10
2*7=14
3*5=15
3*7=21
5*7=35

How to proceed further?

Thanks
Nandish

NandishSS wrote:Set S consists of all prime integers less than 10. If two numbers are chosen form set S at random, what is the probability that the product of these numbers will be greater than the product of the numbers which were not chosen?

A. 1/3
B. 2/3
C. 1/2
D. 7/10
E. 4/5

It is not possible for the two products -- that of the two chosen numbers and that of the two unchosen numbers -- to be equal.
Each of the two products has the same probability of being greater.
Thus, the probability that the product of the two chosen numbers is greater = 1/2.

The correct answer is C.

Alternate approach:

The four prime numbers less than 10 -- 2, 3, 5, 7 -- can be divided into pairs as follows:
chosen = 2*3, unchosen = 5*7
chosen = 2*5, unchosen = 3*7
chosen = 2*7, unchosen = 3*5
chosen = 5*7, unchosen = 2*3
chosen = 3*7, unchosen = 2*5
chosen = 3*5, unchosen = 2*7
Of the 6 possible cases, only the last 3 are such that the two chosen numbers have a greater product than the two unchosen numbers.
Thus:
The probability that the 2 chosen numbers have a greater product = 3/6 = 1/2.