Given two decks shuffled decks of 54 cards, split each in half (27 cards), then take one half of each deck and form a new deck of 54 cards. Then shuffle it and pick first two cards. Whats the probability of a match?
I already know how to solve this problem using conditional probability, which is indeed simpler.
However, I want to develop some intuition on how to solve this problem using law of total probability. I added below the solution to this problem using conditional probability, but I tried to also solve using law of total probability but was unable to came up with a full solution.
The real question here is: how to solve this using law of total probability?
My attempts so far use the following logic: the probability of a match depend on how many pairs of cards are formed. So I tried to calculate each probability of exact number of pairs for each k=1, 2, ..., n/2 pairs, and then for each of those probabilities I multiply by the probability of picking a matching pair having k pairs in the deck. However the final probability does not seem to agree with my simulation.
The solution of this problem is easier if we use conditional probability as follows:
Start by observing that a match is only possible if the both come from different decks, as one deck has no repeated cards.
Define de event ${A: \text{both pick cards came from different decks}}$
Then find the probability of both cards coming from different decks. Notice that there is probability of 1/2 of taking the card from a specific half deck, since a card can come from each of the halves with equal probability.
Below $V_1$ means the first valued picked, and $V_2$ means second value picked.
$P(A) = P(V_1 \text{from deck 1})P(V_2 \text{from deck 2}) + P(V_2 \text{from deck 1})P(V_1 \text{from deck 2}) = 2 \frac{1}{4} = \frac{1}{2}$
Now $P(V_1 = V_2| A) = 1/54$, then,
$P(V_2 = V_2) = P(V_1 = V_2|A)P(A) = 1/108$
How can we go about solving this without conditional probability?
Thanks
Edit: turns out this calculation using conditional probability is wrong, as the probability of cards coming from different decks is not 1/2. The probability of taking a card from, say, deck 1, changes when you pick a card from deck 2.
Correct probability is 1/106.
Here's how you can solve this using the total law of probability.
Let $E$ be the event of a match, and let $X$ denote the number of matches in your newly formed deck of $54$ cards.
For $n\in \{0,1,2,\dots ,27\}$ we have $$P(X=n)=\frac{{54 \choose n}{54 - n \choose 27 - n} {27 \choose 27-n} }{{54 \choose 27}^2}$$ $$P(E|X=n)=\frac{n}{{54 \choose 2}}$$ From the total law of probability, $$P(E)=\sum_{n=0}^{27}P(E|X=n)P(X=n)=\frac{1}{106}$$