The chance of a successful event after N failed trials

probabilitystatistics

Forgive me if this has been answered, but I couldn't quite find what I'm looking for. Maybe I'm not used to the mathematical terms.

Say there is an event with a chance of 1 in 20 of being successful. This chance doesn't change in each trial.

If there were N trials (let's say 2 for a concrete example), then what is the chance of the next trial (N+1) being successful?

Trial 1: Fail
Trial 2: Fail
Trial 3: ?

Best Answer

This depends on how your trials are set up. Specifically whether they're independent or not. Look at two cases:

  1. Each trial is throwing a 20-sided die, and success is if you get a 20. In this case trial 3 will still have a 1 in 20 probability of success, no matter what happened in trial 1 or 2. This goes against the intuition that many people have ("surely after 30 fails I must be getting close"), but that is entirely false. The die does not remember and does not keep track.
  2. You have picked out 19 black playing cards and one red, and shuffled teen together. Each trial is picking the top card, looking at it, and then throwing it away. Success is if you draw the red card. In this case, failure on the first 2 trials means you have a 1 in 18 chance of success on the third trial.

(There is a 1.5 where you picked out 19000000 black cards and 1000000 cards.) Note that in both cases, the probability of success on any specific trial is 1 in 20. But in the second case knowledge of the result of other trials changes the probability.

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