Poisson Probability Distribution

 In a shop, the customer arrives at a mean rate of $2$ per min. Find the probability(up to three decimal places) of arrival of $6$ customers in $1$ minute using the Poisson distribution formula.

The number of car accidents along a certain stretch of road occurred at mean rate of $5$ per week. After the introduction of speed cameras the number of accidents in one week was $2$ . Assuming that the number of accidents can be modelled as a Poisson distribution, test at the $5\%$ nominal significance level if there has been a reduction in the number of accidents.

Calculate the following probability, using a suitable approximation where appropriate.

$P(X<3)\text{ given that }X~B(100,0.02)\text{. }$

The annual number of deaths nationally from a rare disease, $X$, may be modelled by the Poisson distribution with mean $25$. One year there are $31$ deaths and it is suggested that the disease is on the increase. What is the probability of $31$ or more deaths in a year, assuming the mean has remained at $25$? Use $\left[P\left(Z<1.1\right)=0.8643\right]$

The annual number of earthquakes registering at least $2.5$ on the Richter Scale and having an epicentre within $40$ miles of downtown Memphis follows a Poisson distribution with mean $6.5$. What is the probability that at least $9$ such earthquakes will strike next year? Use $\left[P\left(Z<0.78\right)=0.7823\right]$

An old university has a high tower that is quite often struck by lightning. Records going back over hundreds of years show that on average the tower is struck on $3.2$ days per year. It is suggested that a likely effect of global warming would be an increase in the number of days on which the tower is struck. The following year the tower is struck by lightning on $7$ days. Carry out a suitable hypothesis test at the $5\%$ significance level.

It is known that nationally one person in a thousand is allergic to a particular chemical used in making a wood preservative. A firm that makes this wood preservative employs $500$ people in one of its factories. What is the probability that more than two people at the factory are allergic to the chemical?

The number of orders placed at an online store is $4500$ per hour. Find the probability of one order occurring per second, if the number of orders placed can be modelled as a Poisson distribution.

Potholes occur independently and at random at a rate of five in a stretch of road $1$ kilometre long. Calculate the probability that there are exactly eight potholes in a randomly chosen stretch of road $2$ kilometres long.

A manufacturer of chocolate bars states that the number of whole hazelnuts in a randomly chosen $100 \mathrm{g}$ hazelnut chocolate bar can be modelled as a random variable having a Poisson distribution with mean $7.2$. Find the probability that in a randomly chosen $100 \mathrm{g}$ hazelnut chocolate bar there are: exactly eight whole hazelnuts. (Correct the answer up to three decimal place)

Calculate: $P\left(X<3\right)$ if $X~P_{\mathit{0}}\left(4\right)$. 

In a particular town, it was found that potholes occur independently and at random at a rate of five in $1$ kilometre stretch of road. The probability that in a randomly chosen $1$ kilometre stretch of road in this town there will be fewer than three potholes is

At a certain company, machine faults occur randomly and at a constant mean rate of $1.5$ per week. Following an overhaul of the machines, the company boss wishes to determine if the mean rate of machine faults has fallen. The number of machine faults recorded over $26$ weeks is $28$. Use a suitable approximation and test at the $5\%$ significance level whether the mean rate has fallen.

The number of misprints per page of a newspaper follows a Poisson distribution with mean two per page. Following new procedures, $49$ misprints are found in $32$ pages of the newspaper.

How many misprints would be needed for a Type I error to have occurred?

At a certain company, machine faults occur randomly and at a constant mean rate of $1.5$ per week. Following an overhaul of the machines, the company boss wishes to determine if the mean rate of machine faults has fallen. The number of machine faults recorded over $26$ weeks is $28$. Use a suitable approximation and test at the $5\%$ significance level whether the mean rate has fallen.

A small shop sells, on average, seven laptops per week. Following a price rise, the number of laptops sold drops to four per week. Test at the $5\%$ significance level whether the sales of laptops have significantly reduced.

The number of calls to a consumer hotline can be modelled by a Poisson distribution with mean $62$ calls every $5$ minutes. Salina believes this average is too low and observes the number of calls recorded during a randomly chosen $5$-minute interval to be $70$. Stating the null and alternative hypotheses, test Salina's belief at the $10\%$ significance level.

The number of errors made by customers when using online banking transactions each week follows a Poisson distribution with mean $25$.

Katya calculates that it is almost certain that the number of customer errors on a randomly chosen day is greater than $11$ and less than $39$. Use calculations to show how Katya arrived at her conclusion.

The number of errors made by customers when using online banking transactions each week follows a Poisson distribution with mean $25$.

Find the probability that there are more than $32$ customer errors in a randomly chosen week.

The number of tea lights that are lit at a place of memorial during one day follows a Poisson distribution with mean $38$. How many tea lights should be available to be at least $98\%$ certain that there are sufficient tea lights for the demand?