Question

The number of accidents in a year attributed to taxi drivers in a locality follows Poisson
distribution with an average 2. Out of 500 taxi drivers of that area, what is the number of
drivers with at least 3 accidents in a year?
(a) 162
(b) 180
(c) 201
(d) 190​

Answer 1

Answer:

For poison distribution P(x=r)=e−m.mrr! where x =0,1,2

Given m=3

i) No.accidents in a year, i.e.r=0

P(x=0)=e−3×(3)00!

P(x=0)=0.0498×11

P(x=0)=0.0498

therefore expected number of drivers with no accidents

=N×P(x=0)

=1000×0.0498

=49.8≈50

ii)More than 3 accidents in a year

P(more than 3 accident)=1−p[x=0,1,2]

=1−[e−3.300!+e−3.311!+e−3.(3)22!]

= 1−[0.0498+0.1494+0.2241]

=1−0.4233

P(more than 3 accident)=0.5767

Expected number of drivers with more then 3 accidents

= N×p(more than 3 accidents)

=1000(0.5767)

=576.7≈=577

Answer 2

Answer:

The number of drivers with at least 3 accidents in a year is 162

Step-by-step explanation:

Given:

Poisson distribution with an Average mean m is 2

Number of taxi driver is 500

To find: the number of drivers with at least 3 accidents in a year.

Solution:

Probability:

• Probability is a method for determining how likely something is to happen.
• Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about how probable an event is to happen, or its chance of happening.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Given that,

X follows the position with mean m = 2 and  n = 500

The probability of at least three accident

P(X≥3) = 1 - P(X < 3)

We know that ,

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

P(X≥3) =  1 - P(X < 3)

          = 1 - [P(0) + P (1) + P (2)]

          = 1 - [ $\frac{e^{-2}2^{0} }{0!}$  + $\frac{e^{-2}2^{1} }{1!}$  + $\frac{e^{-2} 2^{2} }{2!}$]

           = 1 - $\frac{1}{(2.71828)^{2} } [1 + 2 + 2]$       where $e^{-2} =0.13534$

           = 1 - 0.6767

           = 0.3233

The probability of at least three accident is 0.3233

Calculate the number of drivers with at least 3 accidents in a year

Number of drivers = N × P(X≥3)

                              = 500 × 0.3233

                             = 161.65

                             = 162

Number of driver with at least 3 accident is 162

Final answer:

The number of drivers with at least 3 accidents in a year is 162

#SPJ2