Question

An eastern airport has recorded a monthly average of five near misses of landings and takeoffs in the past 5 years. We assume that the distribution of this number is Poisson. a. Find the probability that during a given month there are no near misses on landings and takeoffs at the airport. b. Find the probability that during a given month there are five near misses. c. Find the probability that there are at least five near misses during a particular month.

Answer 1

The answer to this question can be defined as follows:

In option a "0.0067".

In option b "0.1755".

In option c "0.5595".

Explanation:

It is about the distribution of Poisson:

`\lambda = 5 (rate)`

t = 1

The formula for calculating Mean:

`\to \mu = \lambda * t`

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

calculate the value of Mean:

`\to \mu =5 * 1\\\\ \to \mu= 5`

In point a:

When the value of x is equal to 0:

`\to P(X = 0) = (e^(-5) 5^(0))/(0!)`

`= (e^(-5) * 1 )/(0)\\\\ = 0.0067 * 1 \\\\ = 0.0067`

In point b:

When the value of x is equal to 5:

`\to P(X = 0) = (e^(-5) 5^(5))/(5!)`

`= (e^(-5) * 3125 )/(120)\\\\ = (0.0067 * 3125)/(120) \\\\ = (21.0560)/(120)\\\\ =0.1754 \\\\`

In point c:

When the value of x is equal to 5:

`\to P(X \ge 5) = 1 - P(X < 5) \\\\ \to P(X \ge 5) = 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3) - P(X = 4) \\\\ \to P(X \ge 5) = 0.5595`