Answer to Question #178609 in Statistics and Probability for Nicolas Flavio

Question

1. The number of accidents recorded on a freeway possesses a Poisson distribution with an average of 

three (3) accidents per week.

a. What is the probability that there will be no accident in a particular week?

b. What is the probability that there will be exactly five (5) accidents in a particular week?

c. Find the expected number of road accidents on the freeway per year if the weekly numbers of 

the recorded accident are independent.

2. A survey unofficially claimed that in every five (5) young executives, only one (1) practices good reading 

habits.

a. What is the probability that out of 10 young executives, two executives practice good reading 

habits?

Answer 1

a. We use Poisson's formula

${P_n}(m) = \frac{{{\lambda ^m}}}{{m!}}{e^{ - \lambda }}$

By condition, $\lambda = 3$ .

Then

$P(0) = \frac{{{3^0}}}{{0!}}{e^{ - 3}} \approx 0.0498$

Answer: $P(0) \approx 0.0498$

b. Likewise

$P(5) = \frac{{{3^5}}}{{5!}}{e^{ - 3}} \approx 0.1$

Answer: $P(5) \approx 0.1$

c. Lets find the expected number of road accidents on the freeway per day:

$\frac{\lambda }{7} = \frac{3}{7}$

Then expected number of road accidents on the freeway per year is

$\frac{3}{7} \cdot 365 \approx 156$

Answer: $\approx 156$

2.. a. The pribability that aexecutive reads habits good:

$p = \frac{1}{5} \Rightarrow q = 1 - p = \frac{4}{5}$

Then, according to the Bernoulli formula, the wanted probability is

${P_{10}}(2) = C_{10}^2{p^2}{q^8} = \frac{{10!}}{{2!8!}}{\left( {\frac{1}{5}} \right)^2}{\left( {\frac{4}{5}} \right)^8} = \frac{{589824}}{{1953155}} = 0.301989888$

Answer: 0.301989888