what distribution can be used to approximate probabilities for x ?

Question

A computer terminal can pick up an erroneous signal from the
keyboard that does not show up on the screen. This creates a silent
error that is difficult to detect. Assume that for a particular keyboard the
probability that this will occur per entry is 1/1000.let x be the number of silent errors in 12000 entries.

 

what distribution can be used to approximate probabilities for x ?use this distribution to find the probability of at least one silent error?

 

could you please explain to me how can I solve it?

Answer 1

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Answer 2

For a fixed value of np, as n goes to infinity, the binomial distribution tends to the Poisson.  If np is large, then the normal distribution can be an approximation.

With this problem, n, 12000, is large, and np = 1/1000 * 12000 = 12

Thus, we approximate this with P(Lambda = 12), which has distribution e^-12 12^x/x!

Then, the probability of at least 1 error is 1 - P(0 errors) = 1 - e^-12 12⁰/0! = 1 - e^-12  = .99999385578.

 

The exact probability, which would come from the binomial distribution, is 1 - P(0) = 1 - .999¹²⁰⁰⁰ = .999993893

Clearly, the Poisson approximation is a god one.