I am learning probability from Scheldon Ross' book.
The question reads like this:
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?
Solution given is:
Each ordering of the 52 cards can be obtained by first ordering the 51 cards different from the ace of spades and then inserting the ace of spades into that ordering. Furthermore, for each of the 51! orderings of the other cards, there is only one place where the ace of spades can be placed so that it follows the first ace. Thus,
$$P{(\text{the ace of spades follows the first ace})} = \frac{51!}{52!}=\frac{1}{52}$$ Similarly, the probability that the two of clubs (or any other card) follows the first ace is also $\frac{1}{52}$.
which sounds reasonable. However the author carefully notes the confusion usually made while solving this problem. I got his point, but somehow the explanation given to eliminate the confusion is still confusing me. The author notes below:
Many people’s common reaction is to suppose initially that it is more likely that the two of clubs follows the first ace, since that first ace might itself be the ace of spades. This reaction is often followed by the realization that the two of clubs might itself appear before the first ace, thus negating its chance of immediately following the first ace.
Upto this I got it, but did not get how the probability of first ace itself is ace of spade and probability that the two of clubs comes before the first ace are same, as author explains:
However, as there is one chance in four that the ace of spades will be the first ace (because all 4 aces are equally likely to be first) and only one chance in five that the two of clubs will appear before the first ace (because each of the set of 5 cards consisting of the two of clubs and the 4 aces is equally likely to be the first of this set to appear), it again appears that the two of clubs is more likely. However, this is not the case as explained above.
I did not get this: only one chance in five that the two of clubs will appear before the first ace (because each of the set of 5 cards consisting of the two of clubs and the 4 aces is equally likely to be the first of this set to appear)
What I think is we have to prove: $$P(\text{first ace itself is an ace of spade}) = P(\text{two of clubs appear before first ace})$$ How authors' explanation proves this. And if it does not, then how it can be proven?