Question

A storm measuring `25m xx 7m` is over a reservoir measuring `3m xx 2m`. The storm produces 12 lightning strikes per minute. What is the probability of the reservoir not being struck in one minute? I get 0.6579 but the textbook answer is 0.5926.

How long would it take before the probability of the reservoir being struck is at least `3/4`?

Answer 1

At a rate of 12 strikes per minute, the probability of no strikes in one minute is about `0.6579.`

It would take 3 min 20 sec (i.e. 40 strikes) before the probability of the reservoir being struck is at least `3/4.`

Explanation:

We can calculate the relative amount of area of the reservoir compared to the area of the whole storm:

`(3xx2)/(25xx7)=6/175`

This is also the probability that any single lightning bolt hits the reservoir.

Let `X` be the number of lightning strikes needed until one hits the reservoir. Then `X" ~ GEO"(p=6/175):`

`"Pr"(X=x)=(1-p)^(x-1)p`

`color(white)("Pr"(X=x))=(169/175)^(x-1)(6/175)`

and

`"Pr"(X <= x)=1-(1-p)^x`

`color(white)("Pr"(X <= x))=1-(169/175)^x`

for `x=1,2,3, etc.`

Given that strikes occur 12 times a minute, we seek the probability that it takes more than 12 strikes `("12 strikes"/"min"xx"1 min")` before one hits the reservoir:

`"Pr"(X>12)=1-"Pr"(X<=12)`

`color(white)("Pr"(X>12))=1-[1-(169/175)^12]`

`color(white)("Pr"(X>12))=(169/175)^12`

`color(white)("Pr"(X>12))~~0.6579`

This is consistent with the answer you got. However, I notice that if the rate of lightning strikes is increased to 15 per minute, we get `P(X>15)~~0.5926,` so the answer key may have used a rate of 15/min.

For part 2, we wish to answer how many lightning strikes (i.e. how many intervals of 5 seconds) it will take before there's a 75% chance the reservoir has been struck.

`"Pr"(X<=x)=1-(169/175)^x=3/4`

`color(white)("Pr"(X<=x)=1-)(169/175)^x=1/4`

`=>x=log_(169//175)(1/4)`
`color(white)(=>x)=log(1/4)/log(169/175)~~39.74`

We round up to get 40 lightning strikes, which gives

`"40 strikes"/"1 strike/5 sec"="200 sec"="3 min 20 sec".`

Again, this is presuming the strike rate is 12/min. At a rate of 15/min, the answer would be 2 min 40 sec.