Exactly. Just like I can either be struck by lightning or not so the odds of being struck by lightning are 1/2. It’s science.
I usually grasp these probability puzzles pretty okay, but this one eludes me.
Now suppose that the older child isn’t a boy born on Tuesday. The younger child then must be, of course. Now we count up the possibilities for the sex and birth day of the older child. If she’s a girl, she might have been born on any day of the week, generating seven more possibilities. If he’s a boy, he could have been born any day except Tuesday. (Otherwise this case would already have been counted in the first scenario: the older child a boy born on Tuesday).
Now maybe I’m taking a math problem and adding too much grammar into it, but does the statement ‘one of whom is a boy born on a Tuesday’ really preclude that the other could also be a boy born on a Tuesday?
Not at all. The circumstances of Child A have absolutely no influence on the circumstances of Child B, therefore there is a 50/50 chance that Child B is a boy, regardless of the gender of Child A.
Quiet, Mighty, I’m asking someone knows what they’re talking about.
tiohn
1666
The preceding paragraph takes care of the cases where both were born on a Tuesday:
If the older child is a boy born on Tuesday, there are 14 equally likely possibilities for the sex and birth day of his younger sibling: a girl born on any of the seven days of the week or a boy born on any of the seven days of the week. (This analysis ignores minor differences like the fact that slightly more babies are born on weekdays than on weekend days.)
I just typed up a big thing and then realized I get it.
I get it everyone! It makes sense now!
Zylon
1668
This is the sort of logic that I reject with extreme prejudice:
Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2.
If the outcome of a probabilistic scenario can be modified by what someone is thinking when they pose it, then something is horribly broken. These people need to come up with a way to encode the “missing” information into the verbal problem statement, or STFU.
This made me lawl. Also, a Keith Devlin is u!
That’s exactly what the author of the problem is trying to say, too! You agree with his logic. No need to reject it. (That’s my reading, anyhow.)
It is always 1/2, the way they state it, in the real world. These problems exist simply to show how selective information can cause the mathematics to skew away from reality, but the reality is always 1/2. It’s a math nerd wank, nothing more.
The interesting example of counterintutive odds is the Let’s Make A Deal scenario, but that’s a two-phase operation, which is necessary to have the first choice influence the second. These child riddles are just mathematicians and statisticians trying to prove the sky isn’t blue. edit: For fun, not because they’re stupid.
H.
tiohn
1672
No, it’s actually not that, Houngan. Problems like this don’t demonstrate mathematics “skew[ing] from reality”, they demonstrate that the reality of certain probabilities aren’t always intuitive. Probability works the same in the real world as it does on paper.
But, as the author of the paper states, these sorts of problems are highly artificial and are constructed precisely to screw with our intuition.
The relavent paragraphs:
The remarkable thing that Foshee’s variation points out is that any piece of information that affects the selection will also affect the probability. If, for example, you selected a family at random among those with two kids, one of whom is a boy who plays the ukulele and wants to become a dancer, the ukulele-playing and dancing ambitions would affect the probabilities about the sex of his sibling.
Peres says that we shouldn’t despair about our probabilistic intuition, as long as we apply it to familiar situations. The difficulty of these problems is rooted in their artificiality: In real life, we almost always know why the information was selected, whereas these problems have been devised to eliminate that knowledge. “The intuition develops,” he points out, “to handle situations that actually occur.”
Zylon
1673
I’m going to have to run some simulations.
Now to figure out how to simulate intent in code.
You first need to code a sentient AI, then! :)
Houngan
1675
But I don’t see how it screws with my intuition when my intuition is correct in saying it remains 1/2. I took the paper as saying that “as we add in more and more variables to obfuscate the original bad premise (that foreknowledge of one thing had something to do with the outcome of another, unrelated thing) we move closer and closer to the correct answer, 1 in 2.”
Ezdaar
1676
I think you can find some nice Monty Hall problem sites which let you run simulations that give some empirical evidence for the somewhat counter-intuitive answer to this type of problem.
tiohn
1677
But your intuition IS wrong. In the case of these problems, the answer isn’t 1/2, and this is because of how and when the random selection is made and much of the subtlety lies in what the actual selection is:
Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2. The child’s sex and birthday are just information offered after the selection is made, which doesn’t affect the probability in the slightest.
Gardner himself tripped up on his simpler Two Children Problem. Initially, he gave the answer as 1/3, but he later realized that the problem is ambiguous in the same way that Peres argues that the Tuesday Birthday Problem is. Suppose that you already knew that Mr. Smith had two children, and then you meet him on the street with a boy he introduces as his son. In that case, the probability the other child is a son would be 1/2, just as intuition suggests. On the other hand, suppose that you are looking for a male beagle puppy. You want a puppy that has been raised with a sibling for good socialization but you are afraid it will be hard to select just a single puppy from a large litter. So you find a breeder who has exactly two pups and call to confirm that at least one is male. Then the probability that the other is male is 1/3.
In the scenario of Mr. Smith, you’re randomly selecting a child from his two children and then noticing his sex. In the puppy scenario, you’re randomly selecting a two-puppy family with at least one male.
The remarkable thing that Foshee’s variation points out is that any piece of information that affects the selection will also affect the probability. If, for example, you selected a family at random among those with two kids, one of whom is a boy who plays the ukulele and wants to become a dancer, the ukulele-playing and dancing ambitions would affect the probabilities about the sex of his sibling.
The exact opposite is true: probability is not very intuitive, and math helps us understand reality when our intuition fails us.
Devlin begins his following column by noting that a bunch of applied statisticians wrote to tell him that his answer was wrong. That doesn’t surprise me, because the logic of this puzzle commonly plays out in applied statistics.
Notice the boy-girl problem has “two phases” just like the Monty Hall problem: in phase one some process selects a child whose gender we learn, and in phase two we assign probabilities to the other child’s gender. Analytically, this looks just like: in phase one some process selects patients to complete randomized trials for a new drug, and in phase two researchers evaluate the probability the drug works. Just like in the boy-girl problem, the answer hinges on how the stochastic processes in the two “phases” interact.
Without the mathematical apparatus to analyze such problems statisticians couldn’t do their jobs, because the math tells us how to interpret the information the real world is spitting as us.
I disagree. The child-gender question is no different than if I toss two coins and one of them shows Heads. What is the probability of the other one coming up Heads?
Regardless of anything else, the probability is always 50%. No matter how many coin flips you make beforehand, no matter what the results of previous flips are, the outcome of any individual flip is always 50%.
Intuition in this case is right.
And if I roll a die then the probability of me getting a six is always 1/6. Therefore the answer to the Tuesday boy problem is 1/6.
