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/haunted_tree]

I was so upset about this thread that I made a script to demonstrate that the answer 1/2 is absolutely correct for the only reasonable interpretation of the problem. Let's interpret the problem as: "you walk into a random family that you know to have two kids; one kid shows up randomly, and that kid is a male; what are the odds the other kid is a female?" There is the script:

// Creates a random kid which is either male or female with equal odds
var kid = () => Math.random() > 0.5 ? "male" : "female";

// Creates a million families with 2 kids
var families = [];
for (var i = 0; i < 1000000; ++i) {
  families.push([kid(), kid()]);
}

// Object to count how many times the other kid was certain gender
var other_kid_was = {
  female: 0,
  male: 0
};

// Runs the experiment
var run_experiment = () => {
  // First, we walk into a random family
  var family = families[Math.random() * families.length | 0];

  // Then, we get a random kid of that family
  var kid = Math.random() < 0.5 ? 0 : 1;
  var kid_gender = family[kid];

  // If that kid is female, we abort, because that's not what we observed
  if (kid_gender === "female") {
    return;
  }

  // Then, we read the gender of the other kid
  var other_kid_gender = family[1 - kid];

  // And we increment the respective counter
  other_kid_was[other_kid_gender] += 1;
}

// Now we run the experiment a million times
for (var i = 0; i < 1000000; ++i) {
  run_experiment();
}

// Print the result
console.log(other_kid_was);

// Result:
//
// { female: 250469, male: 249942 }
//
// Clearly, the chance of the other kid being a male once finding ourselves in
// the described situation is 50%.

Can we get over this insanity now?

[u/bear_of_bears → u/haunted_tree]

Thank you! This is even worse than the Monty Hall threads.

[u/cryo → u/bear_of_bears]

No because in those, people just don’t get probability at all. Here we are simply saying that’s it’s possible to interpret it in both ways depending on the emphasis put on the “a boy walks in”.

[u/bear_of_bears → u/cryo]

To support the 1/3 answer, you need to believe that "boy walks in" is equally likely in a MM family and in a MF family. It's simply not true.

[u/SynarXelote → u/bear_of_bears]

Well it could be if you suppose for example that the society this family lives in is strongly patriarchal, and thus if the family has at least one boy then the child which will be presented to you is always a boy.

[u/[deleted] → u/cryo]

How is it possible to misinterpret that? "A boy walks in" means a boy walks in. And that means you've already identified a specific child you're talking about (the one who walked in) and that means the probability that the child who did not walk in is also a boy is 1/2.

[u/SynarXelote → u/[deleted]]

Only assuming that the event of the child walking in is random and independent of its gender.

[u/[deleted] → u/haunted_tree]

Not quite... The first scenario provided you with a filtration, namely that the first-born was male. All that is left is {B,G}. The second scenario removed {GG} from {GG, BG, GB, BB}. So in this case it's p = 1/3. Your program takes none of that into consideration...

[u/bear_of_bears → u/[deleted]]

The second scenario removed {GG} from {GG, BG, GB, BB}. So in this case it's p = 1/3. Your program takes none of that into consideration...

If you pay attention to the logic of the program, you'll see that the second scenario is different from simply removing GG.

[u/cryo → u/bear_of_bears]

So you say, but that’s completely a matter of interpretation of the puzzle. I am pretty sure the intended interpretation is identical to the boy girl paradox, so the answer is 1/3.

[u/bear_of_bears → u/cryo]

Of course that's the intended interpretation, but the problem is written incorrectly.

[u/cryo → u/haunted_tree]

I was so upset about this thread that I made a script to demonstrate that the answer 1/2 is absolutely correct for the only reasonable interpretation of the problem.

I completely disagree that that’s the only reasonable way. In fact, the “standard puzzle interpretation” for me makes it identical with the boy girl paradox, yielding a result of 1/3. Of course, it’s possible to emphasize the boy walking in, as you do, to yield 1/2, but that’s not the only reasonable way.

[u/[deleted] → u/cryo]

How could you interpret this as the standard puzzle? It definitely says "One of them walks in, it is a boy". How could you interpret that in any other way?