Average number of trials until success with rising probability












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I'm struggling to solve following problem:



Given the base success chance $P_0$ = $s$ and increasing probability of success each failed trial (in my case assume simple constant increment $d$: $P_n$ = $s$ + $n$*$d$), find average expected number of trials $E(n)$.



Probability of success at $n$-th trial could be expressed as $(1 - P_1)*(1 - P_2)*..*P_n$ (Think of all the previous attempts being negative since we are looking for first successful trial). However, I'm not sure how to find $E(n)$ besides enumerating all possibilities (highly unwanted!).



In addition, how would the solution change for generalized model with function $D(n) = P_n$ expressing probability of each trial? (Let $D(n)$ be geometric progression or $sin(n*Pi/6)$ haha)










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    I'm struggling to solve following problem:



    Given the base success chance $P_0$ = $s$ and increasing probability of success each failed trial (in my case assume simple constant increment $d$: $P_n$ = $s$ + $n$*$d$), find average expected number of trials $E(n)$.



    Probability of success at $n$-th trial could be expressed as $(1 - P_1)*(1 - P_2)*..*P_n$ (Think of all the previous attempts being negative since we are looking for first successful trial). However, I'm not sure how to find $E(n)$ besides enumerating all possibilities (highly unwanted!).



    In addition, how would the solution change for generalized model with function $D(n) = P_n$ expressing probability of each trial? (Let $D(n)$ be geometric progression or $sin(n*Pi/6)$ haha)










    share|cite|improve this question



























      0












      0








      0







      I'm struggling to solve following problem:



      Given the base success chance $P_0$ = $s$ and increasing probability of success each failed trial (in my case assume simple constant increment $d$: $P_n$ = $s$ + $n$*$d$), find average expected number of trials $E(n)$.



      Probability of success at $n$-th trial could be expressed as $(1 - P_1)*(1 - P_2)*..*P_n$ (Think of all the previous attempts being negative since we are looking for first successful trial). However, I'm not sure how to find $E(n)$ besides enumerating all possibilities (highly unwanted!).



      In addition, how would the solution change for generalized model with function $D(n) = P_n$ expressing probability of each trial? (Let $D(n)$ be geometric progression or $sin(n*Pi/6)$ haha)










      share|cite|improve this question















      I'm struggling to solve following problem:



      Given the base success chance $P_0$ = $s$ and increasing probability of success each failed trial (in my case assume simple constant increment $d$: $P_n$ = $s$ + $n$*$d$), find average expected number of trials $E(n)$.



      Probability of success at $n$-th trial could be expressed as $(1 - P_1)*(1 - P_2)*..*P_n$ (Think of all the previous attempts being negative since we are looking for first successful trial). However, I'm not sure how to find $E(n)$ besides enumerating all possibilities (highly unwanted!).



      In addition, how would the solution change for generalized model with function $D(n) = P_n$ expressing probability of each trial? (Let $D(n)$ be geometric progression or $sin(n*Pi/6)$ haha)







      probability statistics expected-value






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      share|cite|improve this question













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      edited Nov 25 at 4:15

























      asked Nov 25 at 4:08









      ewigetraumzeit

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