Question

In a city with 40000 people, each person has a 0.00020 probability of committing a crime each hour. What is the probability that there are exactly 3 crimes during the next hour

Answer 1

Answer:

2.86% probability that there are exactly 3 crimes during the next hour

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

$P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}$

In which

x is the number of sucesses

e = 2.71828 is the Euler number

$\mu$ is the mean in the given time interval.

In a city with 40000 people, each person has a 0.00020 probability of committing a crime each hour.

So $\mu = 40000*0.0002 = 8$

What is the probability that there are exactly 3 crimes during the next hour

This is P(X = 3).

$P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}$

$P(X = 3) = \frac{e^{-8}*(8)^{3}}{(3)!} = 0.0286$

2.86% probability that there are exactly 3 crimes during the next hour

Answer 2

Answer:

Probability that there are exactly 3 crimes during the next hour is 0.0286.

Step-by-step explanation:

We are given that in a city with 40000 people, each person has a 0.00020 probability of committing a crime each hour.

Firstly, we consider the above situation as of Poisson distribution;

The probability distribution of Poisson distribution is given as;

$P(X=x) = \frac{e^{-\lambda} \times \lambda^{x} }{x!} ; x = 0,1,2,3,....$

where, $\lambda$ = rate at which crime is being committed each hour

So, $\lambda$ = $40000 \times 0.00020$ = 8

Let X = No. of crimes during the next hour

So, X ~ Poisson($\lambda = 8$)

Now, probability that there are exactly 3 crimes during the next hour is given by = P(X = 3)

     P(X = 3) = $\frac{e^{-8} \times 8^{3} }{3!}$ = $\frac{e^{-8} \times 512 }{6}$

                   = 0.0286

Therefore, probability that there are exactly 3 crimes during the next hour is 0.0286.