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)
probability statistics expected-value
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
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add a comment |
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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)
probability statistics expected-value
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
$endgroup$
add a comment |
$begingroup$
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
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
$endgroup$
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
probability statistics expected-value
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
edited Nov 25 '18 at 4:15
ewigetraumzeit
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
asked Nov 25 '18 at 4:08
ewigetraumzeitewigetraumzeit
12
12
add a comment |
add a comment |
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