Problem Solving

Brent@GMATPrepNow wrote:Source: Magoosh Practice Questions

A bag contains 5 white marbles and 5 black marbles. If each of 5 girls and 5 boys randomly selects and keeps a marble, what is the probability that all of the girls select the same colored marble?
a) 1/126
b) 1/120
c) 1/24
d) 4/25
e) 1/2

Thought I'd repost the question here for reference.

winniethepooh wrote:Can the question not be solved in this way ?

The first girl can choose any marble in 10 ways
second girl in 4 ways
third girl in 3 ways
forth girl in 2 ways
and fifth in 1 ways

Coming to boys:
The first can select in 5 ways
second in 4 ways
third in 3 ways
fourth in 2 ways
and fifth in 1 way

I just need to know what number should this whole thing be divided by?

I understand accepting new ways to solve problems, but somwthing we've already learn should also work right?

Hi Winneiethepooh,

You're on the right track with this. Here you are using counting techniques to answer the question.

In your solution, you are essentially treating each ball as a distinct (unique) object, where selecting one white marble is different from selecting a different white marble.

So, we can consider the marbles as W1, W2, W3, W4, W5, B1, B2, B3, B4, B5, where selecting B1 is different from selecting B2.

Okay, so P(girls get same color) = [# of outcomes where girls get same color] / [total # of outcomes]

You have already calculated the number of outcomes where girls get same color.
This equals 10*4*3*2*1*5*4*3*2*1

Now we need to determined the total number of possible outcomes.
Well, the first child can select a marble in 10 ways
The next child can select a marble in 9 ways (since the first child already took a marble)
The next child can select a marble in 8 ways
The next child can select a marble in 7 ways
etc.
So the total number of possible outcomes = 10!

This means that P(girls get same color) = [10*4*3*2*1*5*4*3*2*1] / [10!]
= 1/126

Tons of work here, but the solution is the same.

Cheers,
Brent

[email protected] wrote:Hi Brent can you please tell me that why did you take "1 and 1 later on in the following equation:

1*(4/9)*(3/8)*(2/7)*(1/6)=1/126

According to me it should be (1/10 * 4/9 * 3/8 * 2/7 * 1/6)
Please tell me why the 1 in the beginning wherein it should be (1/10) right as the girl will pick any ball initially.

Hi Amit,

You're missing something from my calculations.
It should be 1 x 4/9 x 3/8 x 2/7 x 1/6 x 1

The first 1 is the probability that the 1st girl selects any marble
The probability that she selects any marble is 1. In other words, we can guarantee that she will definitely select any marble.

The last 1 is the probability that the boys get same color marble from remaining marbles.
In our calculations, we are assuming that all of the girls choose the same colored marble. So, once the girls have selected their marbles, the 5 remaining marbles must be the same color.
If the 5 remaining marbles are the same color, the boys are guaranteed to all get the same color.
So, the probability that the boys get same colored marble = 1

I hope that helps.

Cheers,
Brent