Yet another (probability)

Originally posted by lemon lime
It would have to be 1/2. It's not the same as the Monty Hall problem, but the part about the students correctly assessing the probability at 2/3 threw me for a loop... I still don't see how that can be correct. Eliminating one of the outcomes will leave only 2 possible outcomes.

You haven't said anything is eliminated. The probability is 1/2 if you count all four
possible outcomes. In addition to double heads or double tails, one head and
one tail can fall two different ways. There is a 1/2 probability that the professor's
first toss will be a head and tail.

Originally posted by HandyAndy
You haven't said anything is eliminated. The probability is 1/2 if you count all four
possible outcomes. In addition to double heads or double tails, one head and
one tail can fall two different ways. There is a 1/2 probability that the professor's
first toss will be a head and tail.

Okay, I see now why it's a parodox. The heads/tails can show up as TH or HT, but regardless of the order you still have only one coin showing heads and one coin showing tails.

Originally posted by lemon lime
Okay, I see now why it's a parodox. The heads/tails can show up as TH or HT, but regardless of the order you still have only one coin showing heads and one coin showing tails.

Where's the paradox?

Originally posted by HandyAndy
Where's the paradox?

Wolfgang called it a paradox...

"The paradox is that we all know it is 1/2.
Yet the students are correctly telling us it is 2/3. "

Originally posted by lemon lime
Wolfgang called it a paradox...

"The paradox is that we all know it is 1/2.
Yet the students are correctly telling us it is 2/3. "

Okay. Before the coins are tossed, the probability is 1/2. But then, after the toss,
the professor eliminates one of the possible outcomes -- double heads. This leaves
three possible outcomes, two of which are head and tail.

Originally posted by HandyAndy
Okay. Before the coins are tossed, the probability is 1/2. But then, after the toss,
the professor eliminates one of the possible outcomes -- double heads. This leaves
three possible outcomes, two of which are head and tail.

But whatever happens - you are told you have a 2/3 probability of a head and tail.
This cannot be true since we know it is 1/2.
Paradox!

Originally posted by wolfgang59
But whatever happens - you are told you have a 2/3 probability of a head and tail.
This cannot be true since we know it is 1/2.
Paradox!

We only believe it is 1/2 because the student is withholding information (viz., the professor doesn't see two heads).

Originally posted by wolfgang59
But whatever happens - you are told you have a 2/3 probability of a head and tail.
This cannot be true since we know it is 1/2.
Paradox!

Why do we "know" it's 1/2?

Originally posted by HandyAndy
Why do we "know" it's 1/2?

LoL
I 'knew' it was 1/2 before reading the explanation, but now I don't know if I actually knew that or not.

Maybe we could call this Schrödingers Pennies... we know the two flips show one head and one tail, but we don't know if it's HT or TH until we see it.

Originally posted by lemon lime
we know the two flips show one head and one tail...

as well as two heads or two tails.

Originally posted by HandyAndy
as well as two heads or two tails.

Yes, but it's only with the head/tail flip there will be two possibilities (according to order). If you put a mark on both sides of one of the pennies then you could also determine order for double heads and tails...

H*H, HH*, T*T, TT*

And if you apply this same ordering to HT TH then you would have...


... a whole lot of T's and H's.

Originally posted by HandyAndy
Why do we "know" it's 1/2?

Because the chance of head and tail is 1/2. (TH or HT from {TH TT HH HT} )

Originally posted by wolfgang59
Because the chance of head and tail is 1/2. (TH or HT from {TH TT HH HT} )

True. But then someone tinkers with the results before we make our wager.

Originally posted by HandyAndy
True. But then someone tinkers with the results before we make our wager.

Paradox or not, both can't be true (1/2 and 2/3) unless the conditions are different.

Originally posted by HandyAndy
True. But then someone tinkers with the results before we make our wager.

There's no tinkering and it's not a paradox. Before one of the double sides is eliminated (from the calculation) there is a 1/2 chance of getting heads/tails.

After one of the double sides is eliminated (from the calculation) there is then a 2/3 chance of getting a heads/tails combination, regardless of the order... doesn't matter if it's HT or TH, because regardless of order you have two possible combinations of H and T, plus one double side combination.


(a single penny flip will always be 1/2 for heads or tails)