Question comes from Introduction to Probability, by J. Blitzstein.
A deck of cards is shuffled well. The cards are dealt one by one,
until the first time an ace appears.Find the probability that no kings, queens, or jacks appear before the
first ace.
Approach 1
Lets simplify the problem by assuming all the cards are dealt.
There are $52!$ possible orderings of the cards, and all of them are equally likely.
Lets fix the first ace to be the ace of spades for now. To precede kings, queens and jacks, the ace can be any of the first $37$ cards. Suppose it is at position $1 \leq i \leq 37$. The $i-1$ cards to the left of the ace, selected from the $36$ valid ones (no kings, no jacks, no queens, and none of the remaining $3$ aces) can be permuted in $(i-1)!$ ways. The remaining $(52-i)$ cards to the right of the ace can be permuted in $(52-i)!$ ways.
Thus, the probability of an orderings where no king, queen, or jack precedes the first ace is
$$\frac{4\sum_{i=1}^{37}\binom{36}{i-1}(i-1)!(52-i)!}{52!}.$$
Note the factor of $4$ that accounts for possible choices of the first ace.
Approach 2
This time, suppose the experiment terminates as soon as the first ace is dealt.
The first ace can be any of the $4$ aces. Once again, it can occur at either of the first $37$ positions. We select the cards to the left of the ace from the $36$ valid options and permute them.
Then, the probability of the desired even it
$$4\sum_{i=1}^{37}\frac{\binom{36}{i-1}(i-1)!}{i!}.$$
Intuition
I really feel like stopping the experiment after an ace is dealt should give us the same probability as dealing all of the cards, since once an ace is observed, the remaining cards are essentially irrelevant in the sense that they won't make the ordering favorable if a king, queen, or jack was observed before the ace and vice versa. However, this intuition is really week. I wouldn't really be able to explain, let alone demonstrate it to someone.
Questions
- Are my calculations in the two approaches correct?
- Is my intuition stated above correct? If so, could you formulate your thoughts as to why that is and show how solutions in the two approaches equate?
Best Answer
The calculation in your second approach is wrong. The numerator correctly calculates the number of ways of dealing $i-1$ cards out of the set of 36 acceptable cards, and then a particular ace as the $i$th card. But the denominator of $i!$ counts the number of ways of dealing a deck of $i$ cards, and that's not what you want. You want the total number of ways of dealing the first $i$ cards from a deck of 52 cards: $$4\sum_{i=1}^{37}\frac{\binom{36}{i-1}(i-1)!}{\binom{52}i i!}.$$
It's now very easy to see that this is equivalent to your first approach, using the identity: $$\binom{52}i = \frac{52!}{(52-i)!i!}$$
It's a little harder to see that they're both equal to the correct answer of 1/4.