0
$\begingroup$

I've been looking all over to find if there's a formula I can use for this, but here is the problem.

I have an event that has a 5% chance of occurring, but each time it fails the chance to succeed goes up by 5%. I know that the chance of It occuring over X attempts is just 1 - (the chances of failure each time multiplied together). My question however, is with a 5% chance that increases upon failure, what are the chances of getting multiple successes over 10 attempts. The base chance is reset upon a success.

Improve this question
$\endgroup$
5
  • 3
    $\begingroup$ Have you left any useful details out? For instance, you describe what happens to the odds of success given an event failure, but what happens to the odds given a success? How do they change with success? Do they remain pinned at the ingoing level, whatever that is -- e.g., given one previous failure, it would be 10%? Do the odds go back to the baseline 5%? $\endgroup$
    – user78229
    Commented Oct 5, 2017 at 17:24
  • $\begingroup$ The odds of sccuess given a success to revert back to the base of 5% yes. $\endgroup$
    – raelice
    Commented Oct 5, 2017 at 19:48
  • 1
    $\begingroup$ I can't quite figure out what you really mean by "fail" and "succeed" compared to "occurring," but regardless, it sounds like you could model your situation using an urn with 20 balls in it. Each time a ball is pulled out, you put one of a specified color back in. $\endgroup$
    – whuber ♦
    Commented Oct 5, 2017 at 20:49
  • 2
    $\begingroup$ As @whuber hints, probability models with amplification of this kind go back about a century. The conventional phrasing is at first sight alien to your problem as it's about taking balls out of a urn and then replacing them, so some translation is needed. You may have access to jstor.org/stable/1297412 which is one way into the literature. Simulations are instructive, as such processes diverge. $\endgroup$
    – Nick Cox
    Commented Oct 6, 2017 at 11:15
  • $\begingroup$ You could have a look at sciencedirect.com/science/article/pii/S0377042714001745 $\endgroup$
    – kjetil b halvorsen ♦
    Commented Oct 6, 2017 at 11:33

1 Answer 1

Reset to default
1
$\begingroup$

This response should be a comment but it's too long for that format. I'm open to suggestions of deleting this response and breaking it down into multiple, more bite-sized comments. Thread participants (i.e., that relatively small and oligarchic set of active, senior CV members), please weigh in on that.

Your question is analogous to the skill vs luck or "hot hands" debate in statistical, probabilistic sports analysis. "Hot hands" refers to the phenomenon whereby athletes are perceived to be making more successful shots in a run or sequence of shots relative to some long run expectation. Gelman had a post about this a year or so ago. In other words and at least locally for a given block, run or sequence of attempts, that run significantly exceeds the global or unconditional expectation of success. Based on such unconditional analyses, it was concluded that "hot hands" phenomenon didn't exist or, if it did, it couldn't be confirmed using classic statistical models. This conclusion has had a big impact on decision theoretics insofar as it confirmed or was consistent with the assumptions of memoryless markov processes based on exponential distributions. Many very senior statisticians and decision theorists have weighed in on this problem, e.g., Tversky.

That you state that the odds of success, given a success, go back to the long run average of 5% is consistent with this classic or "commonsense" approach to thinking about the "hot hands" issue.

However, there is another way to frame the analysis which amounts to a "success breeds success" point of view, kind of like a statistical Matthew Effect as described by Derek de Sola Price aka Price's Law. By framing the problem as a conditional expectation, the classic, unconditional approach is falsified as incorrect. In other words, given a success, a conditional model contends that the likelihood or odds that the next attempt is also a success goes up. Based on their analysis of the NBA three-point basketball contest, Miller and Sanjuro in a paper titled Is it a Fallacy to Believe in the Hot Hand in the NBA Three-Point Contest? (ungated copy here ... http://www.igier.unibocconi.it/files/548.pdf) develop the likelihood function associated with this phenomenon and demonstrate that, in fact, the "hot hands" phenomenon does exist.

As long as you maintain your statement that, given a success, the odds or likelihood of the next attempt also being a success roll back to the unconditional expectation of 5%, the insights Miller and Sanjuro bring can't help in answering your question.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you so much for the informative response, I will be busy reading all the literature that has been linked in this discussion. $\endgroup$
    – raelice
    Commented Oct 6, 2017 at 17:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.