# Math Help - Bernoulli Trial help

1. ## Bernoulli Trial help

hey guys i have my final coming up and am really struggling. anyone care to help with this problem?

A probability experiment consists of conducting independent Bernoulli trials with probability of success 0.32 until the first success occurs. The expected number of failures preceding the first success has expected value

2. ## Re: Bernoulli Trial help

Originally Posted by skyhawk714
hey guys i have my final coming up and am really struggling. anyone care to help with this problem?

A probability experiment consists of conducting independent Bernoulli trials with probability of success 0.32 until the first success occurs. The expected number of failures preceding the first success has expected value
Often it does happen that in [apparently...] difficult probability problems the solution is surprisingly simple!... in Bernoulli trials the probability to first succes at the n-th trial is...

$p_{n}=p\ (1-p)^{n-1}$ (1)

From (1) You derive that...

$E\{n\}= \sum_{n=1}^{\infty} n\ (1-p)^{n-1}\ p= p\ \sum_{n=1}^{\infty} n\ (1-p)^{n-1}$ (2)

Now You set $1-p=x$ and You obtain...

$E\{n\}= (1-x)\ \sum_{n=1}^{\infty} n\ x^{n-1} = (1-x)\ \frac{d}{d x} \frac{1}{1-x}= \frac{1}{1-x}=\frac{1}{p}$ (3)

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