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I assume there are various ways of explaining it, but what's the most intuitive (yes I understand that is a relative concept)

Edit due to request:

Obvious example: There is a 1/6 chance of rolling a 6 on a 6-sided die. Expected number of rolls is 6/1.

Another example: You have 4/6 chance of rolling 1-4 on a 6 sided die. Expected number of rolls is 6/4.

Another example: There is a 2/52 chance of picking a red queen from a deck of cards. Expected number of draws (with replacement) is 52/2.

edit: Please replace "odds" in my title to "probability". As pointed out in a reply, I did not use the right word.

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  • $\begingroup$ Could you please describe a little more precisely what "complete something" means? What is your experiment, data collection process, or model? $\endgroup$ – whuber Mar 8 at 17:04
  • $\begingroup$ Are my examples enough or do you want something different/additional? $\endgroup$ – lewikee Mar 8 at 17:48
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Let $p$ be the probability of the event (so for you, $\frac{m}{n}$), and $C$ be the expected number of tries,

Then $C = p + (1-p)(C+1)$, since with probability $p$ you get it on the first try, and with probability $1-p$ you don't, and it takes $C+1$ total tries in expectation.

Then we solve for $C$, to get $$C = \frac{1}{p}$$

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  • $\begingroup$ To look at it that way almost seems like cheating! $\endgroup$ – lewikee Mar 8 at 21:12
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I think that the most intuitive explanation is to simply note that a probability of m/n means that out of n trials, the expected number of successes will be m. So if there are n/m trials, the expected number of successes will be 1. This comes down to the fact that the expected number of trials to get k successes is the same as the number of trials to get k expected successes, which is quite intuitive.

P.S. This answer is based on the assumption that the probability of success is m/n. Remember that "probability" refers to successes divided by trials, while "odds" refers to successes divided by failures. So "odds" of m/n means that there are m successes for every n failures, or a probability of m/(m+n).

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