probability

Question

A deck of 52 playing cards is shuffled, and the cards are turned up one at a time until the first ace appears. Is the next card—that is, the card following the first ace—more likely to be the ace of spades or the two of clubs?

Comments

options?

Answer 1

Total ways to arrange a deck of 52 cards  = $52!$

Total ways when ace of spades appears just after ace = $51!$

Total ways when two of clubs appears just after  two of clubs = $51!$

Probability of both the scenarios =  $\large \frac{51!}{52!} = \frac{1}{52}$

Note : Probability of 3 Aces(other than spades) to appear after ace of spades = Probability of 3 Aces(other than spades) to appear after 2 of clubs. So it has no impact on the probability we are calculating.

Comments

MK utkarsh

Total ways when ace of spades appears just after ace = 51!

could you explain how to get 51! ways , please? I have tried but still not getting it.

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Swapnil permute all cards other than ace of spades, $51!$

whichever be the first ace out of 3 ace's left in pack of $51$ cards, place the ace of spades just after that

So there are $51!$ orderings where ace of spades appears just after first ace