
Title:
0542 Tuesday Solution

Description:

If I go ahead and execute this and print the result,

it comes out: 13/27.

Wow! Where did that come from?

So that' surprisingfirst of all, it's not 1/3,

which you might have thought should be the answer

if you believe the argument that Tuesday doesn't matter.

And secondly, not only is it not 1/3,

but it's much closer to 1/2 than it is to 1/3.

So just having the birthday there really changed things a lot.

How did that happen? Well, I wrote up a little function here

to report my findings, and here's its arguments.

You can give it a bunch of cases that you care about

the predicate that you care about

and whether you want the results to be verbose or not.

And it just prints out some information

and, by the way, as part of this,

I also looked at the question of

what's the probability of two_boys,

given that there's one boy born in December

so I threw that in as well.

And here's the output I get: 2 boys, given 1 boy is 1/3;

2 boys, given at least 1 boy born on any day is still 1/3;

and born on Tuesday is 13/27,

and born in December is 23/47.

Now, I can turn on the verbose option to report

In that case, here's what I see:

The probability of 2 boys, given at least 1 boy

born on Tuesdayis 13/27.

And here's the reasonat least 1 boy, born on Tuesday,

has 27 elementsand there they are

and of these, 13 are 2 boysand there they are.

And so, you can't really argue with that.

You can go through and you can make sure that that's correct,

and you can look at the other elements of the sample space

and say no, we didn't miss any

so that's got to be the right answer.

It's not quite intuitive yet, and

I'd like to define my report function

so that it gives me that intuition

but right now, I don't have the right visualization.

So I've got to do some of the work myself.

And here's what I came up with:

We still have the four possibilities that we showed before

but now we're interested, not just in boys

we're interested in boys born on Tuesday.

So there's going to be some others over here

where there's, say, boy born on Wednesday,

along with some other partner

maybe a boy born on Saturday.

But we're not even considering them; we're throwing all those out.

We're just considering the ones that match here.

And like before, we draw 2 circles:

one of the righthand side of the event

of the conditional probability.

And so how many of those are there?

Well, there's 7 possibilities here

because the boy has to be born on Tuesday

there's only 1 way to do thatbut there's 7 ways for the girls to be born.

So there's 7 elements of the sample state there;

likewise, 7 elements over here.

Now how many elements over here?

Well here, either one of the 2 can be a boy born on Tuesday.

So really, we should draw this state

as either a boy born on Tuesday, followed by another boy

or a boy, followed by a boy born on Tuesday.

And how many of those are there? Well, there's 7 of these

by the same argument we used in the other case,

and of these, there's also 7

but now I've doublecounted because in one of these 14 cases

is a boy born on Tuesday, followed by a boy born on Tuesday.

So I'll just count 6 here.

And so now it should be clear: 7, 14, 21, 6, 27.

There's 27 on the righthand side,

and then what's the probability of 2 boys,

given this event of at least 1 boy born on Tuesday?

Well, 2 boysthat's hereso it's 13 out of the 27.

So that's the result.

Seems hard to argue with.

Both the drawing it out with a pen

and the computing worked out to the same answer.

Now why is it that we have a strong intuition

that, knowing the boy born on Tuesday

shouldn't make any difference?

I think the answer is because we're associating

that fact with an individual boy.

We're like taking that fact

and nailing it on to himand it's true.

If we did that, that wouldn't make any difference.

But, in this situation, that's not what we're doing.

We're not saying anything about any individual boy.

If we did that, the computation wouldn't change.

Rather, we're making this assertion

that at least one was born on Tuesday

not about boys, but about pairs.

And we just don't have very good intuitions

about what it means to say something

about a pair of people,

rather than about an individual person,

and that's what we did here

and that's why the answer comes out to 13/27.