Probabilities and Problem Abilities (Part 2 of 3)

We're in the middle of a three-part post on simple probabilities. If you haven't, please see yesterday's to get up to speed.

How many ways can you make a five kid family?
First order of business: Remove head from gutter.
Second order of business: Make sure 1/32 is correct.
Let’s see if there really are 32 possible outcomes when a couple has five kids. And let's see if whether having all boys or all girls is a 1/32 shot.

Key:  b = boy; g = girl; Inside the parentheses = (total boys, total girls)

Five in a row
ggggg (5g, 0b)
bbbbb (5b, 0g)

Four in a row
gbbbb (4b, 1g)
bgggg (4g, 1b)

ggggb (4g, 1b)
bbbbg (4b, 1g)

Three in a row
ggbbb (3b, 2g)
bbggg (3g, 2b)

gggbb (3g, 2b)
bbbgg (3b, 2g)

gbggg (4g, 1b)
bgbbb (4b, 1g)

gggbg (4g, 1b)
bbbgb (4b, 1g)

bgggb (3g, 2b)
gbbbg (3b, 2g)

Two in a row, twice
gbbgg (3g, 2b)
bggbb (3b, 2g)

bbgbb (4b, 1g)
ggbgg (4g, 1b)

bbggb (3b, 2g)
ggbbg (3g, 2b)

Two in a row
ggbgb (3g, 2b)
bbgbg (3b, 2g)

gbbgb (3b, 2g)
bggbg (3g, 2b)

gbgbb (3b, 2g)
bgbgg (3g, 2b)

bgbbg  (3b, 2g)
gbggb (3g, 2b)

Always alternating
bgbgb (3b, 2g)
gbgbg (3g, 2b)

Count all those up and there really are 32 possible series, or possible outcomes, when a couple has five kids.

However...

Independent probabilities
Does this mean that if you’ve had four girls and you’re pregnant for a fifth time that you have 1/32 odds (3% chance) that it will be a girl? Afterall, having five girls has only a 1/32 chance to it. So does being pregnant after four girls mean that you're probably going to have a boy? Are the odds are on a boy's side?

Of course not.

Each babymaking event is independent. Each is 50/50 odds of boy or girl. Even after four girls you have a 50% chance of having another girl. Just like every single other time you made a girl before.

Think about that if you're really really in the mood for a boy after four girls. Are you still interested in having a fifth kid that still has 50% chance of not being the boy of your dreams?

Look. Lots of people know this already and very well, but that doesn't stop them from being seduced by trends, streaks, runs. This is momentum or luck that is most probably not real, yet it can be so very intoxicating in a casino. By having a fifth pregnancy just because you want a boy, you are playing the odds--the same odds you've had each prior birth.

One in 32 has absolutely no meaning to you unless you’re thinking of your family in greater context, in terms of all the families that have ever had five or more children and what their outcomes were.

The odds that you had five girls out of five were 1/32 but so were the odds that you had girl, boy, girl, boy, girl!

Going from series of births to sex ratios in families
Calculating the odds of having x girls and y boys for a total of five kids (x + y = 5) is a different estimate indeed. One way to consider this estimate is that there are six possible boy/girl ratios in a five-child family...

1. 5 boys, 0 girls
2. 5 girls, 0 girls
3. 4 boys, 1 girl
4. 4 girls, 1 boy
5. 3 boys, 2 girls
6. 3 girls, 2 boys

So that's 1/6 odds, or 17% chance, that if you have a family of five children that they'll be all boys. These are the same exact odds that you'll have any other combination of boys and girls totaling five children.

But wait.

It's obvious that there are more ways to get to some boy/girl ratios than others. There is only one way to make a family of five children with five girls (5g, 0b). But there is more than one way to make a family of five children with three girls (3g, 2b). Let's add those series up and get some different answers with this different line of thinking...

To be continued Thursday... (linked here)

And it's already written, so don't nobody go and post the answers below in the comments, okay?