Probability GMAT Prep

[suyash.tiwari]

A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?

A) 1/4
B) 3/8
C) 1/2
D) 5/8
E) 3/4

I have a problem with the solution of this problem.
Why is order considered here ? We just want to know the probability of getting the sum odd.
Thus we have to either get all three numbers odd or (one odd and two even.) Why is ordering put into picture in the second case is beyond my understanding, especially when selection is done with replacement.


Can somebody please help ?

[jlucero]

Great question! In this particular scenario, it would not matter and you could still get an answer of 1/2, but it's important you understand why this isn't the case every time. First off, let's start with this question: because you could get an odd number by adding three odd numbers or by adding two even numbers and one odd, are the odds of these two things happening the same? No. Let's look at the different possibilities:

EEE
OOE
OEO
EOO

(4 scenarios that would add to even)

OOO
EEO
EOE
OEE

(4 scenarios that would add to odd)

Notice that you are three times more likely to choose one even number and two odd numbers than to choose three odd numbers. You are also three times more likely to choose one odd number and two even numbers than to choose three even numbers.

Overall, you have a probability of:

1/8- 3 odds
3/8- 2 odds
3/8- 1 odd
1/8- 0 odds

You DO need to account for the different orders, because if there is more than one way to order a scenario, it is more likely to happen. For example, the odds of getting 1, 2, or 3 odd numbers is more than just 1/4 because getting 2 odds or 1 odd is more likely than getting 3 odds or 0 odds.

Here's an analogous situation: imagine someone walked into a store and said he or she has a 50/50 chance of winning the lottery with this ticket: I could either win or not win, you would say that person is crazy. Because there are a lot of ways to NOT win, the odds of not winning the lottery are better. That's why we have to find the various ways that odds and evens could show up in a probability question.

[suyash.tiwari]

I stumbled while reading your starting text. I think you wrote in a hurry and made mistakes here. Thus I found it hard to get what you meant.

EEE
EEO
EOE
OEE

(4 scenarios that would add to even)

OOO
OOE
OEO
EOO

(4 scenarios that would add to odd)

Can you please correct your post so that I can understand your point.

[jlucero]

I did switch the scenarios, sorry about that. These should be correct now.