# Saturday Puzzle #16 – Pink or Blue?

For the past 63 years in a row, babies born in the US have been slightly more likely to be a boy than a girl, at a rate of roughly 51% to 49% (source). For today’s puzzle, let’s ignore that complication and assume the odds of being born a boy or a girl are precisely 50%. Now, imagine you have four children and consider these three possible outcomes:

1. all four children have the same gender (four boys or four girls)
2. three have the same gender and one has the opposite gender (three girls and a boy or three boys and a girl)
3. an even split (two boys and two girls)

Here’s today’s challenge: tell me which of the three scenarios above is the most likely one. If you’re not sure, just take a guess. Leave me a comment with your answer. Extra credit will be awarded if you can tell me the probability of the most likely outcome.

I myself come from a family of four children – my three siblings and I are pictured above.

Solution: The first step is to understand how many possible permutations we’re dealing with. We have four kids and each one has two possible states (male or female) so that gives us 2 to the 4th, which is 2*2*2*2 = 16 permutations. Another way of seeing that is to list the possible gender configurations. We can do this by starting with four girls, which can happen in only one way (GGGG), then listing the configurations with three girls (GGGB, GGBG, GBGG, BGGG), then those with two girls and so on (I’ve noted in parentheses which category each element in the list belongs to):

1. GGGG (case 1)
2. GGGB (case 2)
3. GGBG (case 2)
4. GBGG (case 2)
5. BGGG (case 2)
6. GGBB (case 3)
7. GBGB (case 3)
8. GBBG (case 3)
9. BGGB (case 3)
10. BGBG (case 3)
11. BBGG (case 3)
12. GBBB (case 2)
13. BGBB (case 2)
14. BBGB (case 2)
15. BBBG (case 2)
16. BBBB (case 1)

Next, we divide those 16 gender configurations into the three categories mentioned in the problem:

Case 1 (4/0): GGGG, BBBB

Case 2 (3/1): GGGB, GGBG, GBGG, BGGG, GBBB, BGBB, BBGB, BBBG

Case 3 (2/2): GGBB, GBGB, GBBG, BGGB, BGBG, BBGG

Finally, we sum the number of permutations in each group and divide by the the total number of possibilities (16) to get the probability of each case:

case 1: 2/16 = 1/8 = 12.5%
case 2: 8/16 = 4/8 = 50.0%
case 3: 6/16 = 3/8 = 37.5%

This result is a bit surprising – because things generally tend to even out over time, most people assume the correct answer is 2/2 but, as you can see, 3/1 is the most likely split. The photo was my way of giving a tiny hint because it depicted the 3/1 (BBBG) split in my own family.

I got several great responses to this one – congrats to my many brilliant readers and my apologies for a little bit of ambiguity in my formulation, which, I think, threw off a few responders. See you next Saturday!

## 9 thoughts on “Saturday Puzzle #16 – Pink or Blue?”

1. Al Pessot says:

The most likely outcome is the 3/1 outcome and it’s 50%
You have 4 children and any one could be boy/girl true/false 0/1,
so there are 16 possible outcomes.
Use the binary representation from 0000 to 1111 and you will see that all 4 the same
only occurs for 0000 and 1111 or 2/16
Even occurs for 0011, 0101,0110,1001,1010 and 1100, there are 6
The remaining 8 values are all of the form 3 0′s and 1 1 or 3 1′s and 1 0
8/16 = 50%

2. Ricardo Agudo says:

Case 2: 1/2 likelihood
Case 3: 3/8 likelihood
Case 1: 1/8 likelihood

3. Dina says:

Here’s my guess: each of the three options are equally likely. Since the outcome of each birth is independent of the previous one(s), I think the probability of any combination of 4 births is 1/16.

p.s. Love that picture!!!

4. Simon "Chaos" Banks says:

This is a point of contention in our house with Morag and myself..
This is the kind of problem you can over-complicate..

50/50 chance either way the average with 4 children is 2 boys and 2 girls..

5. Dina says:

I have to correct my response…the possibility of any specific PERMUTATION is 1/16 (I think). But you’re comparing the probability of combinations. So my new answer is #2.

#1 probability is 2/24
#2 probability is 8/24
#3 probability is 7/24

6. John Baldi says:

By enumerating every possible outcome, there are 16: 1 = all 4 boys, 1 = all 4 girls. There are 4 out of 16 ways to get 3 boys + 1 girl, and 4 out of 16 ways to get 1 boy + 3 girls, and lastly, the most likely, is 6 out of 16 ways to get 2 of each, which is a probability of 37.5%

7. Katy says:

#3 seems like it should be right but I’m coming up with #2 (3 of same gender & one of opposite) when I write it out. So I’ll say #2 and throw common sense out the window (a risky move, I know!)

8. Katy says:

The probability of the most likely outcome being: .5

9. Jim Goss says:

Option 2 most likely 1/2 or 50% chance
Option 1 has a 1/8 chance
Option 3 has 3/8 chance