Question

The number of accidents in a year attributed to a taxi driver in a city follows Poisson distribution with mean 3. Out of 1000 taxi drivers, the approximate number of drivers with no accident in a year given that ${e^{ - 3}} = 0.0498$ is
A.4.98
B.49.8
C.498
D.4.8

Answer 1

Here, we have to use the concept of the Poisson distribution to find the number of drivers with no accident in a year. So we will use the formula of the probability of Poisson distribution and by using the value of the mean given we will find its value for no car accident and then we will multiply it with the numbers of taxi drivers to get the approximate number of drivers with no accident in a year.

Formula used: Probability of Poisson distribution,$P\left( x \right) = \dfrac{{{e^{ - \lambda }}{\lambda ^x}}}{{x!}}$
Where, $\lambda$ is the mean and x is the actual number of successes.

Complete step-by-step answer:
It is given that the mean of the Poisson distribution of the number of accidents in a year attributed to a taxi driver in a city is 3 i.e.$\lambda = 3$.
Now we will use the formula of the probability for the Poisson distribution with mean $\lambda$ to find the numbers of taxi drivers to get the approximate number of drivers with no accidents in a year. So, we get
Probability for the zero accident to occur, $P\left( 0 \right) = \dfrac{{{e^{ - \lambda }}{\lambda ^x}}}{{x!}} = \dfrac{{{e^{ - 3}}{3^0}}}{{0!}}$
Now we will solve the above equation, we get
$P\left( 0 \right) = \dfrac{{{e^{ - 3}}{3^0}}}{{0!}} = \dfrac{{{e^{ - 3}}1}}{1} = {e^{ - 3}}$
It is given that the value of the ${e^{ - 3}}$ is $0.0498$. Therefore
$P\left( 0 \right) = 0.0498$
Now, we have to find the approximate number of drivers with no accident in a year out of the 1000 taxi drivers. So, we will multiply the Probability for the zero accident to occur with 1000.
Approximate number of drivers with no accident in a year out of the 1000 taxi drivers $= 0.0498 \times 1000 = 49.8$
Hence, option B is the correct option.

Note: Statistics is the science of collecting some data in the form of the number and studying it to forecast or predict its future possibility like it was asked in our question. In these types of examples we have to note that we should do the calculation for at least one decimal place.
Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
Mode is the most common or most repeating numbers in the given data.
Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.