I have a probability problem, which I need to simulate in a reasonable amount of time. In simplified form, I have 30 unfair coins each with a different known probability. I then want to ask things like “what is the probability that exactly 12 will be heads?”, or “what is the probability that AT LEAST 5 will be tails?”.
I know basic probability theory, so I know I can enumerate all (30 choose x) possibilities, but that’s not particularly scalable. The worst case (30 choose 15) has over 150 million combinations. Is there a better way to approach this problem from a computational standpoint?
Any help is greatly appreciated, thanks! 🙂
You can use a dynamic programming approach.
For example, to calculate the probability of 12 heads out of 30 coins, let P(n, k) be the probability that there’s k heads from the first n coins.
Then P(n, k) = p_n * P(n – 1, k – 1) + (1 – p_n) * P(n – 1, k)
(here p_i is the probability the i’th coin is heads).
You can now use this relation in a dynamic programming algorithm. Have a vector of 13 probabilities (that represent P(n – 1, i) for i in 0..12). Build a new vector of 13 for P(n, i) using the above recurrence relation. Repeat until n = 30. Of course, you start with the vector (1, 0, 0, 0, …) for n=0 (since with no coins, you’re sure to get no heads).
The worst case using this algorithm is O(n^2) rather than exponential.