Respuesta :
Answer:
(a) The probability is 0.6514
(b) The probability is 0.7769
Step-by-step explanation:
If the number of accidents occur according to a poisson process, the probability that x accidents occurs on a given day is:
[tex]P(x)=\frac{e^{-at}*(at)^{x} }{x!}[/tex]
Where a is the mean number of accidents per day and t is the number of days.
So, for part (a), a is equal to 3/7 and t is equal to 1 day, because there is a rate of 3 accidents every 7 days.
Then, the probability that a given day has no accidents is calculated as:
[tex]P(x)=\frac{e^{-3/7}*(3/7)^{x}}{x!}[/tex]
[tex]P(0)=\frac{e^{-3/7}*(3/7)^{0}}{0!}=0.6514[/tex]
On the other hand the probability that February has at least one accident with a personal injury is calculated as:
P(x≥1)=1 - P(0)
Where P(0) is calculated as:
[tex]P(x)=\frac{e^{-at}*(at)^{x} }{x!}[/tex]
Where a is equivalent to (3/7)(1/8) because that is the mean number of accidents with personal injury per day, and t is equal to 28 because 4 weeks has 28 days, so:
[tex]P(x)=\frac{e^{-(3/7)(1/8)(28)}*((3/7)(1/8)(28))^{x}}{x!}[/tex]
[tex]P(0)=\frac{e^{-(3/7)(1/8)(28)}*((3/7)(1/8)(28))^{0}}{0!}=0.2231[/tex]
Finally, P(x≥1) is:
P(x≥1) = 1 - 0.2231 = 0.7769
