Mathematics, 19.03.2020 20:40 aaronpmoore1010

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?

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Using the Poisson distribution, and considering that intervals are independent, it is found that there is a 0.0067 = 0.67% probability that a particular square inch is not hit by any drops during a given 10-second period.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:

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

The parameters are:

x is the number of successes e = 2.71828 is the Euler number $\mu$ is the mean in the given interval.

In this problem, the rate is of 30 drops per minute, hence, for 10 seconds, considering intervals are independent, the mean is:

$\mu = 10\frac{30}{60} = 10(0.5) = 5$

The probability that a particular square inch is not hit by any drops during a given 10-second period is P(X = 0), hence:

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

$P(X = 0) = \frac{e^{-5}5^{0}}{(0)!} = 0.0067$

0.0067 = 0.67% probability that a particular square inch is not hit by any drops during a given 10-second period.

A similar problem is given at link

Answer from: skatingby8910

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 $e^{-5a}$ .

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Answer from: Quest

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Suppose rain is falling at an average rate of 30 drops per square inch per minute. What is the chanc...