$What are the odds of drawing the two tarot cards that I wanted, then reshuffling the pack, and drawing them both again straight away, in the same order?$

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$What are the odds of drawing the two tarot cards that I wanted, then reshuffling the pack, and drawing them both again straight away, in the same order?$

There were two specific tarot cards in a 78 card pack that I wanted to draw. I drew two cards, and indeed was lucky enough to get the very two cards I had 'wanted'. Are the odds of drawing those 1 in 78 x 1 in 77 = 1 in 6,006? What then are the odds of putting them back in the pack, shuffling the pack and drawing them again the very next time, in the same order, again 'wanting' those exact same two cards again, so that the two draws are somehow linked rather than random? Is that 1 in 6,006*6,006 = 1 in 36,072,036? Or is there a *2 for drawing the pair in the same order? = 1 in 72,144,072?

or is it:

(1 in 78)*(1 in 77) + (1 in 78)*(1 in 77) * 2 = 1 in 24,024

Probability

Benabongo

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We'll assume that the decks are properly shuffled before each draw and that each card is unique. The key info is whether or not it matters what order you draw the two cards in the first draw. Suppose we call the two cards you want A and B. If you only care about drawing one order, e.g. AB, then the probability of drawing that on the first draw is $(1/78)(1/77) = 1/6006$. If you don't care about the order of the first draw than there is an extra factor of 2 to include both AB and BA, which has a probability of $(2/78)(1/77) = 1/3003$.

Now once the order is fixed after the first draw (since you say the second draw is in the same order of the first), the probability of the second draw is $(1/78)(1/77) = 1/6006$.

To get the full probability we take the product of the two. So if the order of the initial draw doesn't matter we get a probability of $(2/78)(1/77)(1/78)(1/77) = 1/18036018$.

However, if you fix which order you want to see the cards before the first draw then the probability is $(1/78)(1/77)(1/78)(1/77) = 1/36072036$.

So your first line of reasoning is in the right direction, the important thing to determine is whether or not the initial ordering of the first draw matters.

Amas

171

Benabongo

0

Thanks so much, One additional question - I drew the cards the very next time after having drawn the previous time, if that makes sense. So if one was to add a probability of that, meaning I would want them to be drawn consecutively, and not after a whole loads of intermediate draws, would that mean multiplying by a factor of 2 again?
- Amas
  
  +1
  
  The probability calculated above assumes we are only considering two draws of two cards, typically this best represents consecutive draws. If you are instead considering the probability of drawing the same two cards with intermediate draws in between, then that would be a separate event.
Benabongo

0

Yes two consecutive draws. Thanks

$What are the odds of drawing the two tarot cards that I wanted, then reshuffling the pack, and drawing them both again straight away, in the same order?$

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