Expected Number of Girls Born in a Hospital on a Given Day

[Stef42]

Homework Statement

The av. birth rate in a hospital is 30 births per day. In a given day 25 boys are born. How many girls do you expect to be born on that day?
(assume the prob. of being a boy is 1/2 and is independent of the total number of births)

Homework Equations

The Attempt at a Solution

I'm bit confused, am I supposed to use binomial/poisson distributions? how am I supposed to find a expected value instead of a probability? Or is it really simple?

thanks for the help.

 

[Senjai]

i haven't done probability myself yet (well its been a few years), but an unrelated tip: the title is a bit misleading, you should name it something along the lines of "Probability problem"

 

[Stef42]

i haven't done probability myself yet (well its been a few years), but an unrelated tip: the title is a bit misleading, you should name it something along the lines of "Probability problem"

thanks for the tip :)

 

[Stef42]

I was thinking that since the probability is 50/50, the number of girls would be 25, but this would mean the total birth rate was 50 in a day, making it much higher than the average. So since the prob of males is independent of the total born, surely the expected number of girls would be 15, as the only piece of information is that the average rate is 30 a day and the probability is 1/2 for either gender, no?

 

[Mark44]

You're misunderstanding the problem. It says that on a given day, 25 girls are born. Since the average birth rate is 30 babies per day, how many girls would you expect to be born?

 

[LearningMath]

Quick correction:

It says that on a given day, 25 girls are born. Since the average birth rate is 30 babies per day, how many girls would you expect to be born?

The OP's question said 25 boys were born on that day:

The av. birth rate in a hospital is 30 births per day. In a given day 25 boys are born. How many girls do you expect to be born on that day?

It might influence the answer - which I don't know.

 

[Mark44]

OK, let me revise one word in what I wrote:
You're misunderstanding the problem. It says that on a given day, 25 boys are born. Since the average birth rate is 30 babies per day, how many girls would you expect to be born?

 

[LearningMath]

Well, the problem definitely confuses me. Here's what I'm thinking, though:

I think I would have expected ~ 15 girls to have been born before the day started.

After hearing about the 25 boys born, then I'd say I'd then expect around 5 girls born, only b/c (I suppose) the fact that the average ratio of boys to girls born (1:1) shouldn't have any effect on the average number of children born in the hospital, which is 30.

So while 25 boys is a lot for one day, it shouldn't really have any bearing on the avg. total number of boys or girls born in that hospital each day, right?

Anyway that's my guess...EDIT: No, I'm sorry - I messed up my numbers - I expect 2.5 girls to be born, since only 5 more can be born b/c of the avg. Kidding, but by my reasoning - which I'm sure is flawed- it should be 2 or 3 if the avg 30 is the constant number (and not the 1/2). Like I said, I'm confused.

It reminds me of this Charles Dicken's comment that I'd read when he said he wasn't going to ride the train anymore b/c the "average annual quota of railroad accidents in Britain had not been filled and therefore further disasters were obviously imminent". I'm probably making the mistake he was thinking of when he said that

 

[LearningMath]

BTW, my other guess is 25.

 

[Mark44]

If 30 babies are born per day on average, and the probabilities for boys and girls are equal, then one would expect 15 girls to be born. Period.

That's all the problem is asking for.

 

[LearningMath]

Ahh, ok. I was way off.

Now, I'm not trying to pick bones - certainly not with any of the posters - but, imo, it's a poorly worded question.

Wouldn't a better way to ask this question be,

"How many girls would you have expected to have been born on that day?"

or

"How many girls did you expect to be born on that day?"

They could also ask, "How many girls do you now expect to be born today?"​

But the way it's phrased, "How many girls do you expect to be born on that day?" after just having given me some relevant information about "that day" is a bad mixture of present tense and past concepts that seems to just be confusing for confusing's sake.

Word choice here is important and really it's a poorly phrased, confusing question.

Sorry, don't mean to whine, just trying to make a point.

Thanks for providing the answer! I was curious..