probability of randomly drawing the same card from 2 decks at the same time? From 54 decks? What is the probability of randomly drawing the same card from 2 decks at the same time?  My math tells me 1:52 but my gut says I have a problem somewhere.
The same card from 54 decks?
Thanks!
 A: With the hidden assumed meaning of random and the hidden assumptions of independence, we are looking for the probability that the two cards drawn are the same. So, it's equal to the sum of all $P_{s,r}$ where $s$ runs over all suits and $r$ runs over all ranks. Since all these $P_{s,r}$ are equal (hidden assumption here), let's say to $p$, we have that the total probability is $p$ times the number of cards, i.e., $52\cdot p$. To find $p=P_{s,r}$ note that the probability of drawing the card $(s,r)$ from each deck is $\frac{1}{52}$ (hidden assumption) and by the (other hidden) assumption of independence, the probability of both cards being $(s,r)$ is $\frac{1}{52}\cdot \frac{1}{52}$. Thus, to conclude, the desired probability is $\frac{1}{52}$. (Intuitively, the choice of the card from the first deck does not matter, once it's chosen the only question is what is the probability that the other card from the second deck will be that card.)
Can you now asnwer the second question?
A: Hints:


*

*How many possible pairs of cards can you draw when you draw 2 cards from 2 decks?

*Of the combinations in point 1, how many pairs will have identical cards?
A: For the second question.
If by "the same card from $54$ decks" you mean one card $54$ times the probability is tiny: first card can be anything; then you have to match it $53$ times, with probability $(1/52)^{53} = 1.2 \times 10^{-91}$.
If you mean "some pair of cards matches" then you're sure to have a match since there are just $52$ possible choices for $54$ cards. This is the pigeonhole principle. For smaller number of decks the probability increases toward $1$ faster than you might think: it's a version of the birthday problem. 
A: Well that's an interesting question. I am not sure this is what you mean, but I will put in more explicit terms the problem you have:


*

*You have 2 decks of cards, each deck having 54 cards, all different
from each other (you know, 2 regular card decks).

*You shuffle both of them.

*You draw one card at a time from both cards, and check if the drawn cards are the same.


Meaning that, after shuffling, you will have two sequences of cards. Let us call those two sequences m and n. So, what we want to know is the probability of
∃x∈{1,2,...,52}: m[x] = n[x]
being true. It translates as: "There exists at least one value of x between 1 and 52, such that the xth element of m is equal to the xth element of n"
So, for a start, pick whatever value of x you'd like. Then, identify what is the value of m[x]. Now, we know there is a $\frac{1}{52}$ chance that n[x] is the same as m[x], which is the same to say that there is a $\frac{51}{52}$ chance that n[x] is not the same as m[x].
There are 52 values of x you could have possibly chosen to analyse. Therefore, there should be a probability of $(\frac{51}{52})^{52}$ that, for all 52 values of x, n[x] is not the same as m[x].
$(\frac{51}{52})^{52}$ is approximately 36.43135%. So if there is a 36.43135% chance that there is not a single pair of equal cards, then there is a 100% - 36.43135% = 63.56865% probability that there is at least one pair of equal cards.
Based on simple logics, people assume the chance that there is at least one pair of equal cards in 2 shuffled decks is significantly low. That's also why people are often surprised when it is empirically shown that such fenomenon happens a lot more frequently than expected.
