Plugging in the duck race numbers, n=6000, k=100, m=3, we get about a 5% chance of winning.
We didn’t win, but on the plus side it was the most rubber ducks I’ve ever seen 🦆

Clearly, what changes here is the number of non-winning combinations. If you buy m tickets, then this number is "n-m choose k". And then we compute the probability of winning as before, although it does not simplify nicely.

Let’s start slow. Suppose there are four tickets (A,B,C,D). But now assume that you bought tickets A and B. There are still six possible combinations of two tickets: AB, AC, AD, BC, BD, CD. But now five of them lead to a prize, hence the probability of winning is 5⁄6.

What if you bought more than one ticket, though? What is the probability of having at least one ticket win? It cannot be simply the sum of probabilities of a single ticket winning.

Applying the formula to the duck race, we get a probability of winning, p = 100⁄6000, about 2%.

Then the number of winning combinations is the difference between the total number of combinations and the number of non-winning combinations. After some algebra, it turns out that the probability of winning is simply k/n.

But what is the number of winning combinations? It seems that it is easier to find the number of non-winning combinations. This is just the total number of combinations of k elements that do not include our ticket: "n-1 choose k".

It is easy to see the pattern. If there are n tickets in total and k randomly chosen tickets win, the total number of possibilities, which is the number of combinations of k elements from the set of n elements, is simply "n choose k".

Now suppose there are 4 tickets (A,B,C,D). Again, you have A, two tickets win.
Possible pairs are: AB, AC, AD, BC, BD, CD.
You win in three pairs, hence the chance of winning is 1/2

Suppose there are 3 tickets (A,B,C). You have A. Two tickets win.
All possible pairs are AB, AC, BC.
You win in two cases, hence the probability of winning is 2/3.

If 1 ticket wins out of 6,000, and you bought 3 tickets, your chance of winning is 3/6000.
But what if 100 tickets win? That’s less obvious. Let's build intuition through simple examples

Thanks so much for the support!

Inspired by this amazing graphic that I found at a ramen place in Brno

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