BTGmoderatorLU wrote: ↑Fri May 19, 2023 11:16 am

**Source: Official Guide**
A couple decides to have \(4\) children. If they succeed in having \(4\) children and each child is equally likely to be a boy or a girl, what is the probability that they will have \(2\) girls and \(2\) boys?

A. \(3/8\)

B. \(1/4\)

C. \(3/16\)

D. \(1/8\)

E. \(1/16\)

The OA is

A

We can solve this question using counting methods.

**P(exactly 2 girls and 2 boys) = (number of 4-baby outcomes with exactly 2 girls and 2 boys)/(TOTAL number of 4-baby outcomes) **
As always, we'll begin with the denominator.

**TOTAL number of 4-baby arrangements**
There are

**2** ways to have the first baby (boy or girl)

There are

**2** ways to have the second baby (boy or girl)

There are

**2** ways to have the third baby (boy or girl)

There are

**2** ways to have the fourth baby (boy or girl)

By the Fundamental Counting Principle (FCP), the total number of 4-baby arrangements = (

**2**)(

**2**)(

**2**)(

**2**) =

**16**
**Number of 4-baby outcomes with exactly 2 girls and 2 boys**
This portion of the question boils down to

**"In how many different ways can we arrange 2 G's and 2 B's (where each G represents a girl, and each B represents a boy)?"**
----------ASIDE-------------------------

When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

**If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....] **
So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:

There are

11 letters in total

There are

4 identical I's

There are

4 identical S's

There are

2 identical P's

So, the total number of possible arrangements =

11!/[(

4!)(

4!)(

2!)]

------------BACK TO THE QUESTION---------------------------

Our goal is to arrange the letters G, G, B, and B

There are

4 letters in total

There are

2 identical G's

There are

2 identical B's

So, the total number of possible arrangements =

4!/[(

2!)(

2!)] =

**6**
So.....

**P(exactly 2 girls and 2 boys)** =

**6**/**16** =

**3/8**
**Answer: A**