Question: 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 exactly 2 girls and 2 boys?

1. \(\frac{3}{8}\)
2. \(\frac{1}{4}\)
3. \(\frac{3}{16}\)
4. \(\frac{1}{8}\)
5. \(\frac{1}{16}\)

Solution and Explanation:
Approach Solution (1):
Number of ways of getting P (GGBB) is\(\frac{4!}{2!{^*}2!}\);
Total number of ways is\(2{^n} = 2{^4}=16\);
\(\frac{6}{16}=\frac{3}{8}\);
We have considered this question to a coin that is flipped for 4 times. What is the probability exactly two heads?
P (all outcomes) =\(\frac{1}{2}*\frac{1}{2}*\frac{1}{2}*\frac{1}{2}=\frac{1}{16}\) ;
P (favorable outcomes) =\(\frac{4!}{2!{^*}2!}=\frac{6}{16}=\frac{3}{8}\) ;

Correct option: A

Approach Solution (2):
The possible combinations will be:
GBGB, GGBB, BBGG, BGBG, GBBG, BGGB
6 possible ways will be there.
Total number of ways is Baby can be a boy or a girl
For each baby, the probability is\(\frac{1}{2}\), for 4 babies, the probability will be\(\frac{1}{16}\)

\(\frac{6}{16}=\frac{3}{8}\)

Correct option: A

Approach Solution (3):
The probability of achieving exactly k successes in n trials is shown below:
Formula: P (Probability of K successes in n trials) =\(^nC_k p^kq^{n-k} \)
n = number of trials
k = number of successes
n – k = number of failures
p = probability of success in one trial
q = 1 – p = probability of failure in one trial
According to the question:
n (4 children) = 4
k (we want exactly 2 girls) = 2
n – k = 2
P (Probability of getting a girl in one trial) =\(\frac{1}{2}\)
q = 1 – p =\(\frac{1}{2}\)

\(^4C_2\frac{1}{2}^2\frac{1}{2}^2=\frac{3}{8} \)

Correct option: A

“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 exactly 2 girls and 2 boys?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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