So, when the problem is presented as "I have two bears, one is a male born on Tuesday, what are the odds the second bear is a male" then that seems to establish "one male born on Tuesday" as a "given", just like the first coin toss in our example.

Now all we have to do is determine the probability of the second bear being male, not of finding two males at least one of which was born on Tuesday.

It would seem that if we wanted to find the second probability, then the problem should be stated differently:

1) "We have two bears. What are the odds that both of them are male and at least one was born on Tuesday?")

or

2) "We have two bears. One is male. What are the odds that the other is also male and that at least one of them was born on a Tuesday?".

I recognize that both of those questions result in different answers, both different from the answer to

3) "We have a two bears, one is a male born on Tuesday, what are the odds that the other bear is also a male?" (which is what I understand the question on the forum board to ask)

So I guess I'm wondering which of these three questions is really being posed because it'seems like it is the third, but the answer more accurately seems to reflect the second.

Why, they are independent regardless of whether the first tos was established or not.

>So, when the problem is presented as "I have two bears, one
>is a male born on Tuesday, what are the odds the second bear
>is a male" then that seems to establish "one male born on
>Tuesday" as a "given", just like the first coin toss in our
>example.

No, we do not know which bear - first or second - is a male born on a Tiesday. "One is a male" is not the same as "First is a male." Perhaps, I should have asked about the other - not the second - bear.

>Now all we have to do is determine the probability of the
>second bear being male, not of finding two males at least
>one of which was born on Tuesday.

Again, we do not know to which bear the condition applies.
>
>It would seem that if we wanted to find the second
>probability, then the problem should be stated differently:
>
>1) "We have two bears. What are the odds that both of them
>are male and at least one was born on Tuesday?")

This is certainly a different problem.

>or
>
>2) "We have two bears. One is male. What are the odds that
>the other is also male and that at least one of them was
>born on a Tuesday?".

This is also a different problem.

>I recognize that both of those questions result in different
>answers, both different from the answer to
>
>3) "We have a two bears, one is a male born on Tuesday, what
>are the odds that the other bear is also a male?" (which is
>what I understand the question on the forum board to ask)

Right. I should be using the word "other" rather than the word "second."

>So I guess I'm wondering which of these three questions is
>really being posed because it'seems like it is the third,
>but the answer more accurately seems to reflect the second.

I may have applied a different reasoning. After counting the number of fm pairs that is 14, assume first that the mail bear born on a Tuesday is the first one (M). Then we have 7 pairs Mm. In the second case, we also have 7 pairs mM. But observe that the pair MM is included in both Mm and mM. So that we have only 13 = 7 + 7 - 1 male pairs.

The table simply helped save a few sentences.

I still don't see how it makes a difference which bear is the "one" and which is "the other" (though I do agree you said "the other" and not "the second" and I realize little things like this often do make a difference so I am sorry for misquoting).

Let's take this premise as an example: "I have two coins, one of them is a coin that I tossed yesterday and it came up heads, what are the odd that the other coin is also heads".

Wouldn't it be safe to assume that it doesn't matter WHEN the established coin was tossed? Isn't it clearly an independant event whether it was tossed yesterday, last Tuesday, five years ago or five seconds ago?

Consider that if the "Male Bear on Tuesday" problem is true as stated on the website, then the probability of "the other bear" being male changes depending on how we measure the birthdate of the bear we know to be male. Consider these variations on this problem:

1) Two bears, one is a male born on Tuesday. What are the odds the other bear is male?
13/27 --> 48.15% (rounded)

2) Two bears, one is a male born in August. What are the odds the other bear is male?
144/564 --> 25.53% (rounded)

3) Two bears, one is a male born on August 22. What are the odds the other bear is male?
133,225/532,535 --> 25.02% (rounded)

4) Two bears, one is a male born on an even numbered day of the month. What are the odds the other bear is male?
12/20 = 3/5 --> 60%

5) Two bears (from another social structure where weeks all have ten days each) and one bear is a male born on Two-sday. What are the odds the other bear is male?
100/390 = 10/39 --> 25.64% (rounded)

(FYI: For these calculations I pretended that the days were evenly distributed among the 12 months and among the even vs. odd days in order to simplify the calculations).

The coin analogy does not spans out. What happened - happened. No question about that.

Mathematically and structurally it should make no difference, right? 50/50 is 50/50 whether it's male/female or heads/tails. You can toss a coin on Tuesday and a bear can be born on Tuesday. No matter the event that occurs on that day, it is still one day out of seven.

The whole bear story only serves as a distraction, I think. If not, please explain the difference. You keep indicating that there is a difference but I just don't see it. Here's the problem re-written word-for-word with the aforementioned substitutions:

"If, say, it is known that one of the coins is heads and was tossed on a Tuesday, what is the probability of the other coin to be also heads?"

With these simple word substitutions would you still stand by the assertion that the probability is 48%?

I disagree with that analogy. It is important that, besides the mail/female property, there is another one - birth day of the week. This is an intrinsic property of a bear/coin; being tossed on Tuesday is extrinsic to the experiment. This is what modifies the probabilities.

To make an analogy with coin tossing, the coins come in two weights - light and heavy - with the probabilities 1/7 and 6/7. It is known that one of the coins came up heads and was light; what is the probability that the other one came up heads?

If you wish you may build up a story. Coins are produced daily. All are identical, except for those that I made on Tuesdays. These ones are lighter. Two coins tossed. One happens to be lighter and show heads. What is the probability that the other one also shows heads?

The first list already contains the possibility that both bears were born on the Tuesday, so it'shouldn't be counted again in the second list; hence 7,7,7,6 rather than 7,7,7,7.

Many thanks.