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Set \(S\) consists of all prime integers less than 10. If a number is selected from set \(S\) at random and then another number, not necessarily different, is selected from set \(S\) at random, what is the probability that the sum of these numbers is odd?

A. \(\frac{1}{8}\) B. \(\frac{1}{6}\) C. \(\frac{3}{8}\) D. \(\frac{1}{2}\) E. \(\frac{5}{8}\)

Set \(S\) consists of all prime integers less than 10. If a number is selected from set \(S\) at random and then another number, not necessarily different, is selected from set \(S\) at random, what is the probability that the sum of these numbers is odd?

A. \(\frac{1}{8}\) B. \(\frac{1}{6}\) C. \(\frac{3}{8}\) D. \(\frac{1}{2}\) E. \(\frac{5}{8}\)

Set \(S =\{2, 3, 5, 7\}\). The question "what is the probability that the sum of these numbers is odd?" is equivalent to the question "what is the probability that one of these numbers is 2 while the other is not?".

\(P(\text{the sum is odd}) = (P(\text{the first number is 2}) * P(\text{the second number is not 2})) +\) \(+ (P(\text{the first number is not 2}) * P(\text{the second number is 2})) =\) \(= \frac{1}{4}*\frac{3}{4} + \frac{3}{4}*\frac{1}{4} = \frac{6}{16} = \frac{3}{8}\).

I quickly listed out the set of numbers (2,2) (2,3) (2,5) (2,7) (3,3) (3,5) (3,7) (5,5) (5,7) and (7,7) and then I counted the only odd pair which left me with a probability of 3/10. Why is this method incorrect?

I quickly listed out the set of numbers (2,2) (2,3) (2,5) (2,7) (3,3) (3,5) (3,7) (5,5) (5,7) and (7,7) and then I counted the only odd pair which left me with a probability of 3/10. Why is this method incorrect?

There are more possibilities when picking two numbers:

This question is basically asking- If two numbers are randomly selected from the set {2,3,5,7}, what is the probability that EXACTLY one of them is 2? (and suddenly it becomes a sub-600 level )

P(Exactly one 2) = [P(2 in first pick) AND P(not 2 in second pick)] OR [P(not 2 in first pick) and P(2 in second pick)] \(=> \frac{1}{4} * \frac{3}{4} + \frac{3}{4} * \frac{1}{4}\) \(=> 2* \frac{3}{16}\) \(=> \frac{3}{8}\)

Option C _________________

One Kudos for an everlasting piece of knowledge is not a bad deal at all...

------------------------------------------------------------------------------------------------------------------------ Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover. -Mark Twain

if a number is drawn then remaining numbers are 3 out of 4

total cases - 4C2*2 ( first selected 02 numbers out of 4 and again as order matters, multiply by 2)= 12

now favorable cases where 2 is always selected- so the combination is (2,3), (2,5), (2,7) again as order matters (3,2), (5,2) (7,2) also possible so total cases = 03+03= 06

probability = 06/12= 1/2

now if you say that if a number is withdrawn and you can draw the same number again for the second chance, hard to understand, until unless it is mentioned that the same number is available in the pool all the time

The sum to be odd, the numbers selected must be either {2,3}, {2,5} or {2,7}.

The probability to drawn the numbers 2 and 3 is 2*\(\frac{1}{4}\)*\(\frac{1}{4}\) (We multiply by 2 because the numbers drawn could be either 2 and 3 OR 3 and 2 => so there are 2 cases AB, BA) The probability is the same for the numbers {2,5} and {2,7}.

So, the probability that the sum of the 4 numbers is odd is: 2*\(\frac{1}{4}\)*\(\frac{1}{4}\)+2*\(\frac{1}{4}\)*\(\frac{1}{4}\)+2*\(\frac{1}{4}\)*\(\frac{1}{4}\)=\(\frac{3}{8}\)

I used the below Proba(sum is odd) = 1 - Proba(sum is even)

The set is made of {2;3;5;7}. Number of all possible events is 4^2, which is 16

The sum is even if and only if you : pick twice an even number : there is only {2;2}, ie. 1 possibility or pick 2 odd numbers out of the 3 in the set: 3^2 ie. 9 possiblities.

Therefore, Proba(sum is even)= (1+9)/16 = 10/16 and hence Proba(sum is odd)= 6/16 = 3/8

Done!!!
_________________

What was previously considered impossible is now obvious reality. In the past, people used to open doors with their hands. Today, doors open "by magic" when people approach them

I think this is a poor-quality question and I don't agree with the explanation. i want to understand. what do you mean by "not necessarily different? that means Set 'S' can have repetition of prime numbers. also, what if set S has (2,2,2,2,3,5,5,7,7) or any combinations of prime numbers less than 10? i marked the answer as a guess.

I think this is a poor-quality question and I don't agree with the explanation. i want to understand. what do you mean by "not necessarily different? that means Set 'S' can have repetition of prime numbers. also, what if set S has (2,2,2,2,3,5,5,7,7) or any combinations of prime numbers less than 10? i marked the answer as a guess.

please clarify

The question is mathematically very precise. It implies that numbers can be repeated. That's why the denominator in the solution is 4 for the second pick too. For example, you can choose 2 and 2 OR 7 and 7. All possible combinations are given here: https://gmatclub.com/forum/m10-183859.html#p1452001 _________________

I think this is a high-quality question and I agree with the solution. I have the same conceptual gap as Demba below. Can you please help?

Usually, for questions like these, I find the total possibilities first and then the favourable outcomes. To do this, its necessary to know if the order of outcome matters.

Clearly, in this question order matters. (2,3) is different from (3,2). How do we understand after reading the question that order matters? Is it because the question says 'not necessarily different' which implies repetition is allowed?

I was comparing this solution with this one - https://gmatclub.com/forum/m20-184246.html#p1859217. In this solution that I pasted, Order does not matter. So, we wanted to know how to decide after reading a question if order matters or no.

I hope I was able to communicate what I intend to say.

Thanking you in advance!

Demba wrote:

I struggle to know when order matters. Why is the answer not simply 1/4*3/4?

Why is picking 2 then 3 or 5 or 7 different from picking picking 3 or 5 or 7 then 2?