In proof testing circuit boards, the probability that any particular diode will fail is 0.01. Suppose a circuit board contains 200 diodes. What is the standard deviation of the number that is expected to fail

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

Solution to the problem

For this case we assume that the random variable of interest X ="number of circuits that are expected to fail in a sample of 200 selected" and we know that:

[tex] X\ sim Bino(n =200, p =0.01)[/tex]

The expected value is given by:

[tex]E(X)= n*p= 200*0.01 = 2[/tex]

The variance is given by:

[tex] Var(X) np(1-p) =200*0.01*(1-0.01) = 1.98[/tex]

And the deviation is given by:

[tex] Sd(X) =\sqrt{1.98}= 1.407[/tex]