While playing solitaire, I wondered, in a shuffled deck, how many cards might you expect to turn over before seeing your first ace? Seems like a simple problem, but it's not, and it leads to some beautiful math, and a surprising use of calculus where you would not expect continuous math at all.

If you want to know how far the last ace is from the end of the deck, the answer is the same; just deal the cards in reverse.

First, a simple case, just 13 cards, ace through king of hearts. Ace could be anywhere from first to last with equal probability, so the average of all of those is 7. You'll see ace on the seventh card, more or less, give or take. However, it's good practice to follow up with a probability tree.

First card ace: 1/13.

Ace as the second card: 2 (position) times 12/13 (not ace the first time around) times 1/12 (ace the second time around).

Third position = 3 times 12/13 times 11/12 times 1/11.

…

Each line collapses to the position number times 1/13. Carry that through and get the sum of (i times 1/13), or 1/13 times ∑ i, which yields 7.

A probability tree can evaluate a traditional deck with 4 suits, but the numbers aren't clean and easy, so best to run it through a computer. If you're on unix, feed this into bc -l. You need -l so bc will do floating point.

suits = 4; t = 0; for(i=1; i<=13*suits - suits + 1; ++i) { s = 1; for(j=1; j<i; ++j) { s = s * (13*suits - suits + 1-j)/(13*suits + 1-j); } s = s * suits / (13*suits + 1-i); t = t + s * i; } t;

Here's the same program written in python.

suits = 4 t = 0.0 for i in xrange(1, 13*suits - suits + 2): s = 1.0 for j in xrange(1, i): s = s * (13*suits - suits + 1-j)/(13*suits + 1-j) s = s * suits / (13*suits + 1-i) t = t + s * i print t

The answer is 10.6. Instead of position 7, your first ace comes along in position 10 or 11.

Now let's extend to an infinite deck, or if you prefer, sampling with replacement. Pull a card out at random and then put it back. Each time the odds of an ace is 1/13, no matter what happened before. How long to an ace? Intuition says 13, since each trial succeeds with probability 1/13. You can see the progression: one suit is 7, 4 suits is 10.6, and an infinite deck is 13. In fact the program has the number of suits at the top. Change it and see what happens. Starting with 1 suit and moving up, you find: 7, 9, 10, 10.6, 11, 11.28, etc, moving inexorably towards 13. (The pattern seems to be 13 - 12/(suits+1), but I'm not gonna try to prove that here!) So let's look at the infinite deck, and that's where calculus comes in.

For notational convenience, say the odds of success is p, and the odds of failure is q, where q = 1-p. In our example, p is 1/13 and q is 12/13. You will find success on your first try with probability p. Second try success is 2 times p times q. Third try success is 3p times q squared. Forth time success is 4p times q3. Take the sum as i runs from 1 to infinity of i times p times qi-1. Pull p out of the sum, but what do we do with the rest? It is ∑ ixi-1, where the real variable x is replaced with q. But aha! ixi-1 is the derivative of xi. So it is the sum of the derivative of xi. Mathematicians have died and gone to heaven so we can exchange summation and differentiation. It becomes the derivative of ∑ xi. If we started i at 0, instead of 1, we would have a perfect geometric series. That series sums to 1/(1-x). Subtract 1, since we are starting at i = 1, and get 1/(1-x) - 1. The derivative is 1/(1-x)2. Substitute q in for x, and remember that 1-q = p, and we have 1/p2. We pulled p out of the sum at the start, so multiply by p and get 1/p. There you have it. The expected number of trials to success is 1 over the probability of success. If the odds of an ace are 1/13 each time, then you can expect to pull 13 cards, on average, give or take, to reach your first ace.

Math is quite lovely, isn't it?