Answer:
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
<u>a) Calculate the probability exactly four thefts occur in a minute.</u>
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
<u>b)What is the probability there is one or less thefts in a minute?</u>
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
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<u>c) What is the probability there are three or more thefts in a minute?</u>
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
