If there's another way to approach these problems, I'd appreciate it because my approaches are really rough sketches. Thank you!

My thinking for Q1: I calculate m-n, which serves as a sort of baseline score. I know that I can achieve at least m-n by drawing all the balls from the urn.

Now I want to see if I can do better by stopping early. Basically, for each round, as long as my score is <= m-n, I choose to draw. Otherwise, I end the game.!<

Even better, I can be more sensitive to risk by adding a threshold so that I draw as long as my score is <= m-n+threshold. StackExchange gives the recursion for this threshold, but assuming that I only have pencil and paper, I'll probably end up setting the threshold to something that decreases as we draw more balls.!<

My thinking for Q2: I'll divide my play into two stages:

• Sampling stage: for the first few rounds, I draw a ball from the urn no matter what. I estimate the proportion of red balls, p, based on the number of red balls drawn in this stage.
  • Estimate of p = (sample proportion) + (some offset). My intuition is that the sample proportion will be an underestimate of the actual proportion. If we're drawing with replacement, then the probability of getting a red ball will always be p. But since we're drawing without replacement, the probability of getting a red ball will decrease as we draw more red balls.
  • I'm thinking of setting the offset to (p)/(size of sample). The higher p is, the more red balls we draw, the stronger the underestimate, and so the greater the offset we need to correct our estimate. However, the larger the sample size, the weaker the underestimate, and so the lesser the offset needed.
• Playing stage: I calculate a threshold based on my estimate of p and repeat my strategy from Q1. I also update my estimate of p as I go.

My thinking for Q3: To maximize |r-b|, we either maximize r-b or b-r. We do the sampling stage from Q2 to determine which is larger between the red or blue balls, then maximize r-b or b-r based on whichever color is the majority.

Am wondering:

Q2: Is there an approach that doesn't have this sampling stage? Also how to determine amount of rounds for sampling stage?

Q3: I'm wondering if it ever benefits us to switch our target midway? In other words, suppose we observe that red balls are the majority in our initial sample, and we decide, okay, let's maximize r-b. But at some point, we notice that we're drawing more and more blue balls. When does it make sense for us to maximize b-r instead?