Question:
You throw 1000 darts. Each one has a 50% chance of scoring. For the first 500 darts, each is worth 1 point, and for the second 500 darts each is worth 3 points. What is the most likely number of 3-point darts you got in given that your score is 1500?
I've tried writing out the probability of getting m 3-point darts in given a score of 1500 but the presence of factorials makes it difficult to find the maximum. I've also thought about using a normal approximation but this also got messy - I'm wondering if there's a simpler way?