Pack of Cards Probability Problem of Gigantic Proportions?

Question

From a well-shuffled pack of cards (whatever that means!), what is the probability of finding JUST two cards, of the same rank (any rank), next to each other?

Or maybe I should have worded it as " what is the probability of finding AT LEAST two cards of the same rank next to each other?"

Either way, I have no idea how to tackle this, but my suspect intuition tells me the probability is > 0.5 from my son's practical experience.
Is this an impossible ask or are my probability skills waning?

Answer 1

Let's calculate an estimate.

Whatever the first card, the prob the 2nd card is of a different rank is 48/51.
Whatever the second card, the prob the 3nd card is of a different rank is 47/50.
You can continue in this mode through the whole pack, but it's not exact, since (for example) you may have had 3 Jacks in the first 10 cards, so the probability of getting a different card after the next Jack will be 1.
But anyway....if we multiply them altogether, as we should, the probability of of having no adjacent cards of the same rank will be roughly (after simplification) 3*2*1/51*50*49 (since all other terms cancel out) which is roughly 1 / 20,000.

Which means the probability that there will be adjacent cards of the same rank is about 0.99995.

So there are a lot of approximations in there, but I suspect this may be roughly correct....
.

Answer 2

This problem is not impossible but is extremely tedious (and therefore prone to error) by any means I know. The probability of only 2 cards of the same rank being adjacent is very, very small, while the probability of at least 2 cards of the same rank being adjacent is likely much greater than 0.5. I agree that a counting program would be in order here.

Answer 3

I don't think that your probability skills are waning, because this is an almost impossible ask ! I have much experience in probability-calculations and you almost have to write a simulation program to calculate this probability !
On second thoughts it's possible, but not easy :
1) Transform the problem to that finding no two cards of the same rank next to each other in a pack of n cards
2) Then you have to calculate the chances that after n cards with the nth card being of rank r, you have seen 0,1,2,3 cards with that rank before
It's feasible, but involves a lot of factorial calculations.