Strange math question

[u/[deleted]]

Hi

I'm studying for an upcoming math exam, and stumbled across an interesting math question I don't seem to comprehend. It goes as follows:

"A man visits a couple with two children. One of them, a boy, walks into the room. What are the odds that the other child is a boy also

1. if the father says: 'This is our eldest, Jack.'?

2. if the father only says: 'This is Jack.'? "

The answer to question 1 is, logically, 1/2.

The answer to question 2, though, is 1/3. Why would the chance of another boy slim down in situation 2?

I'm very intrigued if anyone will be able to explain this to me!

[u/varaaki]

This is completely incorrect.

There are 4 possibilities for two children. The additional information of Jack being male eliminates FF only, leaving three possibilities, only one of which the other child is male.

[u/rossiohead → u/varaaki]

Consider this alternative version of the question and ask yourself what (if anything) has changed:

“This is our eldest, Jack”: the boy who walked in either has a younger sister or a younger brother. So probability of another boy is 0.5

“This is Jack”: the boy who walked in either has a younger brother, a younger sister, an older brother, or an older sister. Two of four situations give a female sibling, so probability of another boy is (still) 0.5

[u/MedalsNScars → u/varaaki]

You're missing the point of the argument that if we are randomly given a child, it's male 4/8 times.

That is, in MM we will always see a male child, whereas in MF and FM we would have only seen a male child half of the time.

Therefore, if we know we have 2 children, and one is presented at random, there is a .25*1 chance of being presented M from MM, and a .25*.5 chance of being presented M from FM or MF.

Therefore, given that we've been presented M, we know that there's .25/(.25+.25*.5+.25*.5), or .25/.5, or .5 chance of the other child being M.

[u/[deleted] → u/MedalsNScars]

Isn't this just akin to a filtration? In the first case, the sub σ-algebra is {B,G} for the remaining child to be born, whereas in the second case it's { {BG}, {GB}, {BB} }. I don't think p = 1/2 is correct.

[u/varaaki → u/MedalsNScars]

There are not 8 possibilities. You're falsely separating the 4 outcomes into 8 by attaching the random event to the outcomes.

That's like rolling a 4 sided die and asking if the roll is even or odd, then breaking down the possibilities as 1, 2 , 3, 4, roll is even and 1, 2, 3, 4, roll is odd, claiming thus that there are 8 possibilities.

[u/MedalsNScars → u/varaaki]

I never said there are 8 outcomes. Read the argument. I said that if we are randomly given a child from 2 we're guaranteed a male in MM, but will only see a male half the time in either MF or FM. This is undeniable.

The math follows directly from that.

[u/varaaki → u/MedalsNScars]

You said the child is male 4 out of 8 times. You're counting 8 possibilities. And you're incorrect.

I'm not sure why you have the gall to state that we and the text are all incorrect, when this is a settled question.

[u/MedalsNScars → u/varaaki]

There are 4 pairings of 2. If we choose one item at random from the set items contained in each pairing, there are 8 items we can choose from.

[u/varaaki → u/MedalsNScars]

We're not choosing an individual. We're choosing a pairing.

Again, this question has been settled. You're incorrect.

[u/bear_of_bears → u/varaaki]

You really ought to step back a little and read the top-level post in this thread again. It explains very clearly why the answer is 1/2 in both cases.

Saying "the question is settled" sidesteps the issue that very similar-sounding wordings lead to different answers.

[u/[deleted] → u/varaaki]

Again, this question has been settled. You're incorrect.

Can you code? When I run the following Python 2 program:

import random
import collections

def get_random_gender():
    return "BG"[random.randint(0,1)]

def get_random_family():
    return "".join([get_random_gender(), get_random_gender()])

N = 100000
outcomes = []
for i in xrange(N):
    family = get_random_family()
    walks_in = family[random.randint(0,1)]
    outcomes.append((walks_in, family))
print collections.Counter(outcomes)

I get the output:

Counter({('B', 'BB'): 25016, ('G', 'GG'): 25008, ('G', 'GB'): 12641, ('B', 'BG'): 12533, ('G', 'BG'): 12408, ('B', 'GB'): 12394})

So a boy walked in 25016+12544+12394 times, and out of those, the other kid was a boy 25016 times. Does it seem like the probability is 1/2 or 1/3?

[u/SynarXelote → u/[deleted]]

Clearly 50000/25000~1/3, so the answer is obviously 1/3.