Question

A type of long-range radio transmits data across the Atlantic Ocean. The number of errors in the transmission during any given amount of time approximately follows a Poisson distribution. The mean number of errors is 2 per hour. (a) What is the probability of having at least 3

Answer 1

32.33% probability of having at least 3 erros in an hour.

Explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\mu` is the mean in the given time interval.

The mean number of errors is 2 per hour.

This means that
`\mu = 2`

(a) What is the probability of having at least 3 errors in an hour?

Either you have 2 or less errors in an hour, or we have at least 3 errors. The sum of the probabilities of these events is decimal 1. So

`P(X \leq 2) + P(X \geq 3) = 1`

We want
`P(X \geq 3)`

So

`P(X \geq 3) = 1 - P(X \leq 2)`

In which

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)`

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

`P(X = 0) = (e^(-2)*(2)^(0))/((0)!) = 0.1353`

`P(X = 1) = (e^(-2)*(2)^(1))/((1)!) = 0.2707`

`P(X = 2) = (e^(-2)*(2)^(2))/((2)!) = 0.2707`

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767`

`P(X \geq 3) = 1 - P(X \leq 2) = 1 - 0.6767 = 0.3233`

32.33% probability of having at least 3 erros in an hour.