The paradox of changing odds

[Bill Stokes]

Three sets of two playing cards - you have:

[A][A] - [A][K] - [K][K]

All 3 sets are placed face down randomly. Your job is to select (guess!) the pair [A][K].

Now, at this stage, the chances of getting the correct pair is 3 to 1 ~ that is not the question.

You select a pair.

I turn over one card of that pair, and it is a [K].

Now what are the chances you have the [A][K] pair? And why, nothing has changed?

Nick

[Parts]

some beans

[Parts]

something has changed....you can now disregard the [A][A], so the odds are 2 - 1

[Bill Stokes]

something has changed....you can now disregard the [A][A], so the odds are 2 - 1

Nothing has changed - why do the odds change just by showing you a card you already have? If I didn't show you one of the cards, are the odds still 3 to 1?

Nick

[Parts]

but you did show me...so the odds go down

[Mr. YOur no fun]

As you don't have knowledge of where the A A combo is that doesn't no good for you in the odds, so I think it is still 3-1.

[Parts]

That's not the point....the AA set is disregarded....you either have AK or KK, so there is a 2-1 chance that you have the AK

[Bill Stokes]

OK, think about this. You choose your two cards. Now then, without looking, the card on the left is either [K] or [A]. That means you either have ( [K][K] or [K][A] ) ~ OR ~ ( [A][K] or [A][A] ).

There, I didn't show you any cards, yet the odds seem to change from [3 to 1] to [2 to 1]. Impossible!

Nick
P.S. I do know the solution and can show the proof.

[Bill Stokes]

Well, seeing as everybody has given up, here is the proof.

The problem is you don't know what [K] (or maybe an [A]) you have, as there are three each.

The possible combinations of the sets of two cards are:

[A1][A2]
[A2][A1]
[K1]K2]
[K2][K1]
[A3][K3]
[K3][A3]

Now, at first sight, and correct, the odds are 3 to 1. Pick one set, and it's a [K] ~ but which one? That means you either have:

[K1][K2]
[K2][K1]
[K3][A3]

...still 3 to 1.

Nick

[Parts]

That's cheating...you added a shit load more cards!

[(29th)JiggyHoot]

That makes sense now. I had to re-read the original topic the key being "3 sets" of cards.

[Bill Stokes]

That's cheating...you added a shit load more cards!

No - there are still only 3 sets of two cards [K1][K2] ; [A1][A2] ; [K3][A3] ;- but do you see [K1] or [K2]?

Nick

[Parts]

yes .... before the cards were [A][A], [K][K] etc, now they all have numbers. but if you have any [K] you can still safely say you don't have the [A][A] set.

I rest my case m'lud

[Bill Stokes]

I rest my case m'lud

Then you are going down for a very long time

You could have either of the [K]'s - you don't know which one. So you still have a one in three chance that you got the set [K][A].

Nick (P.S., this isn't my paradox - it's well known in the maths circles).

[Parts]

Let's look at your 1st post....

Three sets of two playing cards - you have:

[A][A] - [A][K] - [K][K] no numbers here...just 3 K's and 3 A's

All 3 sets are placed face down randomly. Your job is to select (guess!) the pair [A][K] ok, I have to find the only set out of the 3 that has 1 of each

Now, at this stage, the chances of getting the correct pair is 3 to 1 ~ that is not the question.

You select a pair.

I turn over one card of that pair, and it is a [K]. I now know that I don't have the only pair of A's and can have either the pair of K's or the set with 1 of each - the set I was asked to guess

Now what are the chances you have the [A][K] pair? 2-1And why, nothing has changed? yes it has, you turned over 1 of my pair and it was a K

Don't trust these mathy people, they want to trick you all the time with their sneaky numbers things.

[Bill Stokes]

From the original question, the odds (probabilty) is 3 to 1. But the chance is 2 to 1 against.

That means you still have a 2 to 1 against chance by being shown one card of the pile you picked.

I know John, it is doing my head in, but the maths is correct.

Links to read:

https://en.wikipedia.org/wiki/Monty_Hall_problem

https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox

Nick