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3 years, 4 months ago
You have two shuffled decks of cards (52 cards each).
You put them next to each other and deal two at a time, one from each deck.
What is the probability that there will be at least one match? That is, the probability that the two cards turned up will be the same at some point in the game?
Jan 5, 2020 at 20:37
Dr Xorile Dr Xorile
22.4k 3 3 gold badges 44 44 silver badges 119 119 bronze badges
The answer is quite simple when you know the concept of a
derangement, since the only time this doesn't happen is if the first deck is a derangement of the second.
In other words, we are looking for
There are online calculators if you want the exact value, which is approximately
Jan 5, 2020 at 21:25
The easiest approach is if we first calculate how often there are no match at all.
If we match a card from deck 1 with one from deck 2, it doesn't matter what
card 1 shows, there are always 51 chance in 52 for it to not
match with the card 2.
Since this happen 52 times in a row,
we have 52 attempts with a probability of 51/52 that all must go through.
The chance that
there are no match is therefor (51/52)^52 =0.3643 or 36.43%
remaining percent must be all the times it does match.
100% - 36.43% = 63.57%
Jan 5, 2020 at 21:08
Bjorn Eriksson Bjorn Eriksson
1,370 1 1 gold badge 7 7 silver badges 14 14 bronze badges
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