I) Given an infinite deck of cards, each distinct, all ranked.
Lay them out in a row, face up, left to right, in order.

Shuffle a second, identical deck. Deal it one card at a
time, face up, directly below the first.

On average, how many pairs match?

II) Given an infinite deck, composed of replicates
of a standard 52 card deck; that is, an infinite number
of copies. Arrange it into separate decks, each ordered
by suit and rank. Lay them out as above, face up, in
a single row.

Shuffle another identical deck. As before, deal it out
below the first, one card at a time.

On average, what fraction of the pairs match?

--
Rich

On Thursday, August 5, 2021 at 8:54:33 PM UTC-4, RichD wrote:
> I) Given an infinite deck of cards, each distinct, all ranked.
> Lay them out in a row, face up, left to right, in order.
>
> Shuffle a second, identical deck. Deal it one card at a
> time, face up, directly below the first.
>
> On average, how many pairs match?

For each card the chance of matching is 1 / N where N is the number of cards in the deck. For any card to match multiply by N, but that's not quite right since it doesn't consider the chances of multiple matches.

> II) Given an infinite deck, composed of replicates
> of a standard 52 card deck; that is, an infinite number
> of copies. Arrange it into separate decks, each ordered
> by suit and rank. Lay them out as above, face up, in
> a single row.
>
> Shuffle another identical deck. As before, deal it out
> below the first, one card at a time.
>
> On average, what fraction of the pairs match?

In this case the chances of a match are always 1/52 since consuming a card still leaves an infinite number of cards. So the total number of matches over the infinite deck is infinity / 52.

--

Rick C.

- Get 1,000 miles of free Supercharging
- Tesla referral code - https://ts.la/richard11209

On August 6, gnuarm.del...@gmail.com wrote:
>> I) Given an infinite deck of cards, each distinct, all ranked.
>> Lay them out in a row, face up, left to right, in order.
>> Shuffle a second, identical deck. Deal it one card at a
>> time, face up, directly below the first.
>> On average, how many pairs match?
>
> For each card the chance of matching is 1 / N where N is the number of
> cards in the deck. For any card to match multiply by N,

Too easy, hey? So the average is N/N = 1 match. Even with an infinite deck!

> but that's not quite right since it doesn't consider the chances of multiple matches.

Good catch, I forgot about multiple matches.
Don't know the answer exactly, but it must involve an infinite series.

>> II) Given an infinite deck, composed of replicates
>> of a standard 52 card deck; that is, an infinite number
>> of copies. Arrange it into separate decks, each ordered
>> by suit and rank. Lay them out as above, face up, in
>> a single row.
>> Shuffle another identical deck. As before, deal it out
>> below the first, one card at a time.
>> On average, what fraction of the pairs match?
>
> In this case the chances of a match are always 1/52 since consuming a
> card still leaves an infinite number of cards.

Right, but again, the correct answer must account for multiples.

--
Rich

On Saturday, August 7, 2021 at 7:12:00 PM UTC-4, RichD wrote:
> On August 6, gnuarm.del...@gmail.com wrote:
> >> I) Given an infinite deck of cards, each distinct, all ranked.
> >> Lay them out in a row, face up, left to right, in order.
> >> Shuffle a second, identical deck. Deal it one card at a
> >> time, face up, directly below the first.
> >> On average, how many pairs match?
> >
> > For each card the chance of matching is 1 / N where N is the number of
> > cards in the deck. For any card to match multiply by N,
> Too easy, hey? So the average is N/N = 1 match. Even with an infinite deck!
> > but that's not quite right since it doesn't consider the chances of multiple matches.
> Good catch, I forgot about multiple matches.
> Don't know the answer exactly, but it must involve an infinite series.
> >> II) Given an infinite deck, composed of replicates
> >> of a standard 52 card deck; that is, an infinite number
> >> of copies. Arrange it into separate decks, each ordered
> >> by suit and rank. Lay them out as above, face up, in
> >> a single row.
> >> Shuffle another identical deck. As before, deal it out
> >> below the first, one card at a time.
> >> On average, what fraction of the pairs match?
> >
> > In this case the chances of a match are always 1/52 since consuming a
> > card still leaves an infinite number of cards.
> Right, but again, the correct answer must account for multiples.

Actually, the multiple match issue is a red herring since we are counting the average number of matches and not the chance of a match. I was not thinking clearly when I wrote that about the first case. The only issue is if the Nth card in deck A matches the Nth card in deck B counting the total number of expected matches over the infinite deck.

--

Rick C.

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