Respuesta :
Answer:
a) 19.85% probability that a total of two people are struck by lightning during first four months of the year.
b) 22.68% probability that the year has 5 good and 7 bad months
Step-by-step explanation:
We are going to use the Poisson distribution and the binomial distribition to solve this question.
Poisson distribution:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
Binomial distribution:
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
a.Find the probability that a total of two people are struck by lightning during first four months of the year.
10 people during a year(12 months).
In 4 months, the mean is [tex]\mu = \frac{10*4}{12} = 3.33[/tex]
This is P(X = 2).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 2) = \frac{e^{-3.33}*(3.33)^{2}}{(2)!} = 0.1985[/tex]
19.85% probability that a total of two people are struck by lightning during first four months of the year.
b.Say that a month is good is no one is struck by lightning, and bad otherwise. Find the probability that the year has 5 good and 7 bad months.
Probability that a month is good.
P(X = 0), Poisson
The mean is [tex]\mu = \frac{10*1}{12} = 0.8333[/tex]
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.8333}*(0.8333)^{0}}{(0)!} = 0.4346[/tex]
Find the probability that the year has 5 good and 7 bad months.
Now we use the binomial distribution, we want P(X = 5) when n = 12, p = 0.4346. So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{12,5}.(0.4346)^{5}.(0.5654)^{7} = 0.2268[/tex]
22.68% probability that the year has 5 good and 7 bad months
