Question

According to an international aviation firm, fatal accidents occur in just around one in every five million flights. With about 100,000 flights per 24 hour day worldwide, this means fatal accidents happen at a rate of 1 per 50 days. This translates to an average of 0.6 accidents per month. Assume that the number of fatal air accidents worldwide follows a Poisson distribution with λ=0.6 per month.

1. Compute the probability that no fatal air accident will happen anywhere in the world during the month of March, 2020.
2. Compute the probability that more than 3 fatal air accidents happen in a calendar month.
3. Assume that fatal air accidents happen independent of one-another. Compute the probability that no fatal air accidents will happen anywhere in the world between March 1 and May 31, 2020.
4. In the year 2018, 13 fatal air accidents were recorded. Was the year 2018 an outlier with respect to fatal air accidents? Compute the probability that a year would have 13 fatal air accidents. Note that your need to convert the monthly Poisson rate of 0.6 accidents to a yearly rate.

Answer 1

1. Let X be the number of fatal air  accidents in a month

X follow Poisson with =0.6

Probability mass function of a Poisson distribution is

Probability that no fatal accidents happen during the month of March 2020 is 0.5488

2.

Probability that 3 fatal accidents happen during the month of March 2020 is 0.0198

3. There are 3 months between March 1 to May 31

We know that Probability that no fatal accidents happen during one month is 0.5488  

As accidents happen independent of one another , that is accidents in one months is independent of accidents in other months.

Probability that no fatal accidents happen between March 1 to May 31 = 0.5488³ = 0.1653

Note : For independent events , probability of joint event= Product of individual probability  of each event .

4.

Let Y be the number of fatal air  accidents in a year

Y follow Poisson with =0.6*12 = 7.2  

Probability that 13 fatal air accidents happen during a year is 0.0168

this probability is very low

Usually if the probability of an observation is less than 0.05 , the observation is considered an outlier .

Thus 2018 is an outlier with respect to fatal air accidents