Question

Suppose rain is falling at an average rate of 30 drops per square inch per minute. What is the chance that a particular square inch is not hit by any drops during a given 10-second period? What assumptions are you making?

Answer 1

Answer:

Therefore the chance that a particular square inch is not by any drops during a given 10 second period is $e^{-5a}$.

Step-by-step explanation:

Poisson distribution:

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

Given that rain is falling at an average rate 30 drops per square inches per minutes.

A reasonable choose of this distribution is Poisson distribution $(\lambda t)$.

Let the area be a square inch.

Here  $\lambda$ = average rate of rain drop × area

           =(30×a)= 30 a      

t = 10 second $= \frac{10}{60} \ minutes$  $=\frac16\ minutes$.

Therefore,

$P(\textrm {no rain drop in }\frac16 \ minutes) =P(X=0)$

                                               $=\frac{e^{-\frac{30a}{6}}.(\frac{30a}{6})^0}{0!}$

                                              $=e^{-5a}$

Therefore the chance that a particular square inch is not by any drops during a given 10 second period is $e^{-5a}$.