We have a thought experiment for my next class where we are attempting the following:
Prove that the probability of obtaining at least r successes in n trials of a Bernoulli trials experiment, where the probability of success on a single trial is p, is given by
(Im using C(n,r) to denote a combination n choose r)
Pr[At least r successes in n trials] = C(n,r)p^r(1-p)^(n-r) + C(n,r+1)p^(r+1)(1-p)^(n-1-r) + ... + C(n,n)p^n(1-p)^0
So far I have not a whole lot. I decided I would try and work with sequences because I like sequences I guess so I have n trials denoted by (T1, T2, ..., Tn) and I am currently thinking about the probability of each trial, so
Pr(Success in T1) = C(1,1)p^1(1-p)^0 = P
and I think
Pr(Success in T2) = C(2,1)p^1(1-p)^1 + C(2,2)p^2(1-p)^0 = 2p(1-p) + p^2 = 2P
but that seems odd to me... So I suppose my question is am I at all going in the right direction here, and could I maybe have a bit of a boot in the right direction if I am not. As a side note, I haven't taken probability in a few years, so I am definitely rusty, and if I made any dumb errors I ask that you please forgive me.
Prove that the probability of obtaining at least r successes in n trials of a Bernoulli trials experiment, where the probability of success on a single trial is p, is given by
(Im using C(n,r) to denote a combination n choose r)
Pr[At least r successes in n trials] = C(n,r)p^r(1-p)^(n-r) + C(n,r+1)p^(r+1)(1-p)^(n-1-r) + ... + C(n,n)p^n(1-p)^0
So far I have not a whole lot. I decided I would try and work with sequences because I like sequences I guess so I have n trials denoted by (T1, T2, ..., Tn) and I am currently thinking about the probability of each trial, so
Pr(Success in T1) = C(1,1)p^1(1-p)^0 = P
and I think
Pr(Success in T2) = C(2,1)p^1(1-p)^1 + C(2,2)p^2(1-p)^0 = 2p(1-p) + p^2 = 2P
but that seems odd to me... So I suppose my question is am I at all going in the right direction here, and could I maybe have a bit of a boot in the right direction if I am not. As a side note, I haven't taken probability in a few years, so I am definitely rusty, and if I made any dumb errors I ask that you please forgive me.
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Just so you know in the literature Bernoulli trials follow a Binomial distribution.
It is not so much that you need probability as that you need to know combinatorics.
It is actually VERY DIFFICULT to find a "short" formula for what you want - see cummulative in the link below. That is why one usually uses tables or a program like GeoGebra.
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For your problem, you need the formula for EXACTLY k success in n trials where p=probability of success. It is Pr(k;n,p)=(n k)•p^k•(1-p)^(n-k), where (n k)=C(n,k) is the binomial coefficient.
There is an explanation for finding this formula below.
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Given this formula and writing in your notation (n k)=C(n,k)
The probability of "at least r successes in n trials" means that you need the sum of all of the EXACTLY k successes for k≥r. We will write this: Pr(r≤k;n,p)
It is not so much that you need probability as that you need to know combinatorics.
It is actually VERY DIFFICULT to find a "short" formula for what you want - see cummulative in the link below. That is why one usually uses tables or a program like GeoGebra.
----
For your problem, you need the formula for EXACTLY k success in n trials where p=probability of success. It is Pr(k;n,p)=(n k)•p^k•(1-p)^(n-k), where (n k)=C(n,k) is the binomial coefficient.
There is an explanation for finding this formula below.
------
Given this formula and writing in your notation (n k)=C(n,k)
The probability of "at least r successes in n trials" means that you need the sum of all of the EXACTLY k successes for k≥r. We will write this: Pr(r≤k;n,p)
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keywords: Experiment,Bernoulli,Trials,Proof,Bernoulli Trials Experiment Proof