r/probabilitytheory

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It’s a categorical random variable. The outcomes are heads and tails, not 1 and 0.

Edit: I in response to the pushback, I look into how Bernoulli conceived it. It was in the context of an experiment (sampling with replacement) to determine the ratio of pebbles in an urn that had only white or black pebbles.

I’m working with a game where, for example, a player has to hit a moving ball with a racket. Every time they successfully hit the ball, the racket becomes smaller, making the next hit progressively more difficult. Once they miss, the racket resets to its original size.

Each round lasts 30 minutes, and I would like to compare performance across rounds.

I also have a chance/control condition where the racket doesn’t move, so some hits can happen simply by chance.

I would like to get a single normalized hit-rate/performance value for each round that tells me whether the player performed well or poorly overall.

Ideally, this value would take into account:
\\\\- The racket getting smaller after each successful hit

\\\\- The racket resetting after a miss

\\\\- The fixed 30-minute duration

\\\\- Different numbers of hits and misses across rounds

\\\\- The chance-level performance from the stationary-racket condition

What would be the best way to calculate this as one normalized value? I’m particularly interested in something that accounts for the increasing difficulty as the racket gets smaller, rather than just calculating hits / total attempts. I have tried to calculate a score using the difference between the actual number of hits and the expected value of hits, but I think it does not take into account the paddle starting from where they last hit the ball (if it did not miss).

Also, if anyone knows of similar studies or experiments that have used a comparable normalized performance/hit-rate metric, I’d really appreciate any references or examples. Thank youu

Since posterior probability is equal to prior probability, when we simplify we get
P(A intersection B) = P(A)*P(B), so why do we need to check the common between both events, ik this really silly and dumb doubt but i'm not getting the logic behind this please help.

Thank you in advance