This was the question asked by tallanvor: "I have 2 children. One is a boy. What are the odds the other is a girl?" He doesn't say that its the first child that is a boy. Only that (at least) one is a boy. If he said the first was a boy, and what are the odds the second is a girl, then of course the answer is 1/2. But the answer to his question is 2/3.
No, based on the question as posed, we know with 100% certainty that one is a boy. 'First' or 'second' is completely irrelevant. The only variable which remains is the unknown gender of the other child, which is 1/2. He left out the all-important 'IF one is a boy' and instead said 'One is a boy'. It may seem like a trivial difference, but it is critical to the resolution of the question. It's an interesting mental exercise though, because it's really more about semantics than math or statistics. It helps to visualize it as a figure of four points of a square: A B C D Two children means only two flips of the coin. Define A = Boy B = Boy C = Girl D = Girl Prior to actually flipping the coin, the POTENTIAL results are as follows: Either A + B (boy/boy) - a 50% chance, multiplied by an additional 50% chance, or half of half, or 25%. Identically, C + D (girl/girl) - same probability as A + B, 25%. The third and fourth combinations are also the same 25% probability - either C + B (first flip is a girl, second flip is a boy) or A + D (first flip is a boy, second flip is a girl). Both of these have a 25% probability as well. Four different possible outcomes, when taking into account the ORDER of the flips, each of the four at 25% probability, adding to a total of 100% probability that one of those four will with certainty be the result. The problem is this: Prior to any flip being made, the odds are all equal. But, as soon as you flip one, and have filled your first slot with either A, B, C or D, then the probability reduces to 50%. But what if you are not flipping *A* coin, but flipping TWO coins, at exactly the same time? This is where it gets interesting. If I flip two coins at the exact same time, then I get the result without changing the odds. And since C + B and A + D are identical in essence, by flipping two coins at the same time, I no longer have four slots to fill, I only have three. I am either going to get one of three potential outcomes: boy/boy, boy/girl, or girl/girl. Remember, by flipping them at the same time, the order is meaningless, and the uncertainty is retained. The odds are 2/3 that at least one of them will be a boy, and the odds are 2/3 that at least one of them will be a girl. BOTH of them have at least 2/3 odds of being at least one of the two. Both of them. And this is why it is a semantics question and not a math or statistics problem. By saying 'One is a boy', you have not 'flipped the coin' simultaneously, which removes the 2/3 probability. You can't have one outcome at 2/3 without the other one also having the same potential 2/3 outcome. If you give certainty to one, it becomes a completely different question - a single flip of the coin - and becomes 1/2 probability. What those who are presenting the question as given are doing is taking the situation of the 2/3 probability for BOTH genders, which requires zero certainty, and turning it into a simple-sounding word problem and erroneously including certainty where there cannot legitimately be. It would be as though you did the following: place a coin in two cups, one coin in each. Shake the cups, then simultaneously slam the cups upside-down onto a table. The coins have now both been flipped, but because the results are hidden, we do not KNOW the results. Now, even though the results are now set, we don't know the results, so all we can do is guess the probability of the outcome. At this point in time, the odds are 2/3 that under each cup, the answer is a girl. But the odds are ALSO 2/3 that under each cup is a boy. So it is a factually true statement that the odds that under Cup B is 2/3 that it is a girl. But it is also factually true that under Cup B is 2/3 that it is a boy. But because we flipped them down at the same time, revealing what is factually under Cup A does not affect the odds of what is under Cup B. So if we peek under Cup A, and we see that it turned up as a boy, then we can truthfully say that the odds of Cup B turning up as a girl are 2/3. Hilariously though, and the part that is conveniently left out, is that it is ALSO 2/3 probability that under Cup B is a BOY. Identical 2/3 odds of either being a girl OR a boy. In other words, 1/2, a 50/50 chance, which gets back again to it being a question of semantics, and not math or statistics.. So it comes full circle. As long as there is no certainty, it is a legitimate 2/3 chance, but by adding certainty, it is reduced to 1/2.
Should be 50% as the second event is independent of the first one. Just like a coin flip, the second attempt will still have a probability of 50%.
Nice job tallanvor. Your question seems to have really stumped people. Your answer is of course correct.
Ok, here's one: If a tree falls in the forest, and there is no one or no thing there to hear it, does it make a sound? Answer: No, it does not.
As a scientist I say: as it fell, it was moving and caused air molecules around it to compress/move as well. In-situ compression and expansion of air leads to sound, ergo the tree made a sound. I don't quite get the philosophical aspect of this thought experiment... what is it supposed to make me think about? Edit: you and durvasa already got it, but I didn't like the way the question was asked either. If it said "at least one is a boy" then I would go to the 2/3 answer. As written, I also instinctively said 1/2.
Basically, can a sound really be called a sound if there is nothing around to experience it. The sheer essence of sound itself is that it is a sense that can only be experienced by the person being present. I, like you, believe that the impact of the action on the molecules around it is enough for the sound to indeed exist, but philosophers are taught to think differently. Or, in most cases, to not think at all.
That's like the age-old question of "If 713 trolled in a thread, but no one read it, did 713 really post something?"... which of course is ancient Chinese philosophy and will stomp even the Dhalai Lama.
As someone alrdy stated, you can go test it with a coin if you wish. flip a coin twice, circle the events where one is heads. so 3 events: H-T T-H H-H Of those events how many times is the other tails? the answer will converge on 2/3rds the more you try.
I feel the question is unambiguous. "One is a boy" is semantically equivalent to "at least one is a boy."
I see. Thanks. Even if I don't like the sound analogy, I guess it is an interesting idea to think about. My initial reaction is that the statement basically says "you can't assume anything". If you see the tree lying on its side, you can't assume that it didn't just grow that way. But man has progressed this far because we've learnt to make educated guesses which I guess could be one idea the statement wants to lead you to. Eh, that's philosophically deep enough for me.
"One is a boy" made me think "the first is a boy", even though you're right that the English actually implies "at least one is a boy". That's probably the crux of the riddle since the rest is math, and I fell for it.
Here's an extension to that problem that I found yesterday: On your travels, three men stand at a fork in the road. You're not sure which fork you need to take, but each of the three men do. One of these people tells the truth, one always lies, and the third tells the truth sometimes and lies the other times. Each of the three men know each of the others, but you don't know who is who. If you could ask only one of the men (chosen at random, since you don't know which man is which) one yes/no question, what question would you ask to determine the road you wish to take? Spoiler [I haven't figured this one out yet, but the solution is here: Solution
Thinking about this further, I now think you were right that it is ambiguous. "One is a boy" can be interpreted as: (1) "Suppose at least one is a boy." or (2) "Suppose child X is a boy, where X is 1 or 2." For case (1), answer is 2/3. For case (2), answer is 1/2.
