Poll: A little math problem
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This thread proves how much people actually do misunderstand probabilities and how they can change with change in knowledge. Of the respondents, 65.5% got the wrong answer. Let me make an attempt to explain why 33% is the correct answer.
At the start of the problem, we have only know that there are two puppies, but we have no idea what their sexes are. So, the possible outcomes are the following:
dog1 dog2
M M
M F
F M
F F
There are 4 possible outcomes. This means that the probability of each outcome is 1/4 or 25%. If we ask what is the probability that both dogs are male, the answer is 25%. This is because only one of the four outcomes is M/M. But, this is the case that we don't know anything at all about the dogs other than the fact we have 2 of them.
Now, let's suppose we are told that dog 1 is a male. This means that we now know the F/M and F/F outcomes did not occur. So now we ask what are the possible outcomes. The possible outcomes are
dog1 dog2
M M
M F
There are only 2 possible outcomes. So the probability of each outcome is 1/2 or 50%. If we are told that dog2 is the one that is male, then we have again 2 possible outcomes:
dog1 dog2
M M
F M
But, look closely at what happened as a result of our change in knowledge. While the first possible outcome remains the same, the second possible outcome is a different configuration from the case when we knew dog1 was the male. Regardless, because there are only 2 possible outcomes now, the probability of each outcome is 50%.
Now, here is where things change. In the problem, we are told that at least one of the dogs is male. But, we don't know which one. Because we don't know which dog is male, we must consider all possible outcomes in which there is a male dog. There are 3 such outcomes:
dog1 dog2
M M
M F
F M
Notice that the number of outcomes has changed to 3 because we don't know which dog is male, only that at least one is male. Because there are now only 3 possible outcomes, thanks to us knowing now that at least one dog is male, the probability of each outcome is 1/3 or 33%. The problem is asking us what is the probability that both dogs turn out to be male when we know that at least one dog is male. We see that only 1 of the three outcomes have both dogs as male. So, the probability that both dogs are male is 33%.
The KEY of this problem is realizing that the probabilities change as a result of our change in knowledge about the situation. Notice carefully how the probability of having 2 male dogs changed as a result in our change in knowledge about the situation. When we don't know anything at all about the two dogs, the probability of 2 males is 25%. When we know that dog1 or dog2 is the one that male, then the probability of 2 males is 50%. When we only know that at least one dog is male(but not which one), then the probability of 2 males is 33%.
At the start of the problem, we have only know that there are two puppies, but we have no idea what their sexes are. So, the possible outcomes are the following:
dog1 dog2
M M
M F
F M
F F
There are 4 possible outcomes. This means that the probability of each outcome is 1/4 or 25%. If we ask what is the probability that both dogs are male, the answer is 25%. This is because only one of the four outcomes is M/M. But, this is the case that we don't know anything at all about the dogs other than the fact we have 2 of them.
Now, let's suppose we are told that dog 1 is a male. This means that we now know the F/M and F/F outcomes did not occur. So now we ask what are the possible outcomes. The possible outcomes are
dog1 dog2
M M
M F
There are only 2 possible outcomes. So the probability of each outcome is 1/2 or 50%. If we are told that dog2 is the one that is male, then we have again 2 possible outcomes:
dog1 dog2
M M
F M
But, look closely at what happened as a result of our change in knowledge. While the first possible outcome remains the same, the second possible outcome is a different configuration from the case when we knew dog1 was the male. Regardless, because there are only 2 possible outcomes now, the probability of each outcome is 50%.
Now, here is where things change. In the problem, we are told that at least one of the dogs is male. But, we don't know which one. Because we don't know which dog is male, we must consider all possible outcomes in which there is a male dog. There are 3 such outcomes:
dog1 dog2
M M
M F
F M
Notice that the number of outcomes has changed to 3 because we don't know which dog is male, only that at least one is male. Because there are now only 3 possible outcomes, thanks to us knowing now that at least one dog is male, the probability of each outcome is 1/3 or 33%. The problem is asking us what is the probability that both dogs turn out to be male when we know that at least one dog is male. We see that only 1 of the three outcomes have both dogs as male. So, the probability that both dogs are male is 33%.
The KEY of this problem is realizing that the probabilities change as a result of our change in knowledge about the situation. Notice carefully how the probability of having 2 male dogs changed as a result in our change in knowledge about the situation. When we don't know anything at all about the two dogs, the probability of 2 males is 25%. When we know that dog1 or dog2 is the one that male, then the probability of 2 males is 50%. When we only know that at least one dog is male(but not which one), then the probability of 2 males is 33%.
How is the probability for the next bet different than the probabilities for the next million? If they're identical bets, they all have the same probabilities. That includes each individual instance. Doing it a million times just assures that your results will likely mirror the probabilities very closely.Cheeze_Pavilion post=18.73797.826840 said:
"What is the probability that the other one is a male?"Cheeze_Pavilion post=18.73797.826840 said:
That is exactly what they did ask.
Hehe, wow, now this is really irrelevant.Cheeze_Pavilion post=18.73797.826880 said:
I concede that the infinities can't be proven to be different sizes. Fortunately, this little math problem is relatively simple and secular, and doesn't concern such elusive concepts as infinity.
Back to the matter at hand, please?
The proportions remain the same.
1 (2 male pair): 2 (1 male 1 female pairs): 1 (2 female pair)
Again, this is the distribution for random pairs of puppies.
Again, the probabilities remain the same for all scenarios in that infinite number of identical scenarios. Repeating the experiment many times just increases how accurately the physical results resemble the probailities.Cheeze_Pavilion post=18.73797.826902 said:
So, that's analogous to "two random puppies," right?Cheeze_Pavilion post=18.73797.826880 said:
So, what's the matrix of permutations?
Not quite. Natural numbers vs. real numbers is countable infinity vs. uncountable infinity. In informal language, one of those infinities is like "infinity times infinity" (basically the idea is that you can squeeze infinite real numbers between any two rational numbers).Cheeze_Pavilion post=18.73797.826880 said:
This would be more like just taking a limit.
-- Alex
Well, this is separate from that. It speaks to how to test the problem empirically if you want.Cheeze_Pavilion post=18.73797.826950 said:
Here's what I'm driving at...
The problem scenario goes like this:
1. Someone makes a pair out of two random puppies.
2. You ask a question about that pair and find out the answer is yes, allowing you to use this new knowledge to adjust your probabilistic guess.
3. You are asked to make a guess about the pair.
In order to "test" something like this, you should copy that general flow. First, make a pair of random puppies. Then ask "Is at least one male?" Then look at whether there are two males in this set.
If you just try any old way to create a pair of puppies, then you're not effectively simulating the problem anymore.
-- Alex
2 is gambler's fallacy. You're saying "this particular pair of puppies contained a male, so I must create an arbitrary system for generating puppies that puts a male in every pair."Cheeze_Pavilion post=18.73797.827117 said:
What you should be doing is starting with a random pair -- because that's how the pair in the problem is actually constructed. Then for each pair you say "Does this pair contain one or more male puppies?" and "Does this pair contain two male puppies?" and you look at how the distributions are aligned.
Otherwise any "experiment" is flawed because you're replacing the original probability distribution of the sets with one of your own devising.
-- Alex
You are, indeed, doin' it wrong. You are assuming that you know which puppy is the known puppy, and which is unknown. Indeed, if the washer told you, "yes, Sparky's a male," the answer would be 50 percent.Cheeze_Pavilion post=18.73797.827117 said:
However, since you don't know, these three possibilities are all equally likely;
Sparky is a male, Othello is a male
Sparky is a male, Othello is a female
Sparky is a female, Othello is a male
You see how knowing one is male, and knowing which one is male are different? These three solutions are all exclusive possibilities, and all are equally likely. With what do you disagree?
You can't describe order from information that we don't have. Saying "the next one" implies that you know the male dog is first. You can't place him first because you don't know who he is. I don't even think the rules of probability problems allow you to change ordering after a problem is already set up. Sparky and Othello have one set order. It doesn't matter if it's Sparky and Othello or Othello and Sparky. Consistency is important.Cheeze_Pavilion post=18.73797.827175 said: