One study showed that in a certain year, airline fatalities occur at the rate of 0.017 deaths per 100 million miles. Find the probability that, during the next 100 million miles of flight, there will be (a) exactly zero deaths. Interpret the results. (b) at least one death. Interpret the results. (c) more than one death. Interpret the results.

(a) P(X = 0) = 93.24%

Which means that during the next 100 million miles of flight. there is 93.24% chance of exactly zero deaths.

(b) P(X ≥ 1) = 6.76%

Which means that during the next 100 million miles of flight. there is 6.76% chance of at least one death.

(c) P(X > 1) = 0.234%

Which means that during the next 100 million miles of flight. there is 0.234% chance of more than one death, which seems to be very unlikely.

Step-by-step explanation:

We are given that airline fatalities occur at the rate of 0.017 deaths per 100 million miles.

We are asked to find the probability that during the next 100 million miles of flight, there will be

(a) exactly zero deaths. Interpret the results.

The problem can be solved using Poisson distribution given by

P(X = x) = e^{-λ}×λ^{x}/x!

Where λ is the rate of  deaths per 100 million miles of flight and it is also known as the decay rate, x is the variable of interest which is zero in this case.

For x = 0 and λ = 0.017

P(X = 0) = e^{-0.07}×0.07^{0}/0!

P(X = 0) = (0.9324×1)/1

P(X = 0) = 0.9324

P(X = 0) = 93.24%

Which means that during the next 100 million miles of flight. there is 93.24% chance of exactly zero deaths.

(b) at least one death. Interpret the results.

At least one death means equal or greater than one

P(X ≥ 1) = 1 - P(X < 1)

But P(X < 1) means P(X = 0) so,

P(X ≥ 1) = 1 - P(X = 0)

We have already calculated P(X = 0) in part (a)

P(X ≥ 1) = 1 - 0.9324

P(X ≥ 1) = 0.0676

P(X ≥ 1) = 6.76%

Which means that during the next 100 million miles of flight. there is 6.76% chance of at least one death.

(c) more than one death. Interpret the results.

More than one death means greater than one death

P(X > 1) = 1 - P(X ≤ 1)

But P(X ≤ 1) means [P(X = 0) + P(X = 1)] so,

P(X > 1) = 1 - [P(X = 0) + P(X = 1)]

For x = 1 and λ = 0.017

P(X = 1) = e^{-0.07}×0.07^{1}/1!

P(X = 1) = 0.9324×0.07/1

P(X = 1) = 0.06526

Finally,

P(X > 1) = 1 - [P(X = 0) + P(X = 1)]

P(X > 1) = 1 - [0.9324 + 0.06526]

P(X > 1) = 1 - 0.99766

P(X > 1) = 0.00234

P(X > 1) = 0.234%

Which means that during the next 100 million miles of flight. there is 0.234% chance of more than one death, which seems to be very unlikely.