Recent crime reports indicate that 3.8 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. Calculate the probability exactly four thefts occur in a minute. (Round your probability to 3 decimal places.) What is the probability there are three or more thefts in a minute? (Round your probability to 3 decimal places.) What is the probability there is one or less thefts in a minute? (Round your probability to 3 decimal places.)

a) The probability that 4 thefts occur in a minute is 0.19

b) The probability there are three or more thefts in a minute is 0.735

c) The probability there is one or less thefts in a minute is 0.106

Step-by-step explanation:

The formula for an event with a Poisson probability distribution is given by:

P (n events in an interval) = λⁿe^ (-λ) / n! where λ is the average number of events in the interval.

In this problem we have λ = 3.8

a) Calculate the probability exactly four thefts occur in a minute.

λ = 3.8 n = 4

P (4 thefts occur) = 3.8⁴e⁻³⁻⁸ / 4! = 4.665/24 = 0.194

The probability that 4 thefts occur in a minute is 0.194

b) What is the probability there is one or less thefts in a minute?

P (0 thefts occur) + P (1 thefts occur) (n = 0, n = 1)

3.8⁰e⁻³⁻⁸/0! + 3.8¹e⁻³⁻⁸/1! = e⁻³⁻⁸ + 3.8 e⁻³⁻⁸ = 0.022 + 0.084 = 0.106

The probability that there is one or less thefts in a minute is 0.106

c) What is the probability there are three or more thefts in a minute?

This is equal to the probability of (1 - the probability of 0, 1 or 2 thefts occurring) but we already know that the probability of one or less thefts occurring is 0.106 so we only need the probability that there are 2 thefts in one minute

P (2 thefts in one minute) = 3.8²e³⁻⁸ / 2! = 14.44 (0.022) / 2 = 0.159

Therefore, the probability that there are three or more thefts in a minute is

1 - (P (0) + P (1) + P (2)) = 1 - (0.022 + 0.084 + 0.159) = 1 - 0.265 = 0.735