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I wonder about the following situations for a while now and can't quite figure out:

  1. You play lottery once, you have a certain probability to win. When you play more than once, your probabilities don't increase because all games are independent from each other. Nevertheless, imagine you play infinite times, somewhen you should win, or not?
  2. Amateur airplanes. I just read an article about the situation that more and more private pilots drop from the sky and someone stated "Sure, the more fly, the more fall down". I would refer to 1) They fly independent from each other, so why should more airplanes fall?
  3. A car on the highway: In principle the same situation. I drive a mile on the highway and everything is fine. No failure or whatever. When I drive 10000 miles the probability that something will break should increase while each mile is independent from the other. But somehow it isn't. Here I can refer to the lifetime of the devices but though, the probability should increase with the distance and nevertheless, each mile is independent from the other.

What exactly am I struggling with? What do I mix? Or what do I not understand?

Ben
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    1) If there is a positive probability $p$ of winning, then in repeated independent games you with eventually win, and the expected wait for the first win is $1/p$ games. 2) If more planes fly independently of one another then the freq of exp'd accidents increases. If too many fly then maybe they interfere with one another so that independence does not hold. 3) With increasing mileage chances of a failure increase, but a reasonable model is that hours are not indep b/c the failure rate increases with wear.// It is best to take one scenario at a time and understand how to model it correctly. – BruceET Jul 13 '20 at 06:58
  • thanks! 2) and 3) make sense but 1) I didn't understand. The wait for the first win is 1/p? Let's say p=0.1, so the wait will be 10? How/why? – Ben Jul 13 '20 at 07:06
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    (1) Look at _geometric_ distribution. Good place to start with my one-at-a time recommendation. – BruceET Jul 13 '20 at 07:08
  • Thanks! The expectation value is the probability, actually. This means I'd have to play lottery a lot to win somewhen.. :) – Ben Jul 13 '20 at 08:14
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    _Reciprocal_ of probability. – BruceET Jul 13 '20 at 08:15

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