- #1

King_Silver

- 83

- 6

## Homework Statement

The number of tornadoes per year, in Georgia, has a Poisson distribution with a mean of 2.4 tornadoes. Calculate the probability that in any given year, there will be:

(i) At most 2 cases.

(ii) At least one case.

(iii) Calculate the probability that there will be exactly 10 tornadoes in the next seven years.

## Homework Equations

λ =

__2.4__

*(Mean)*,

Formula:

**P(X = x) = e-λ (λx/x!)**

Where X = number of events at given internal

e = ~2.71

x = 0,1,2,3,4……..n

*(where n = any number)*

## The Attempt at a Solution

(i)At most 2, Therefore we need to examine

*P(X=0),P(X=1),P(X=2)*

P(X=0) = e-2.4(2.40/0!) = 0.090717953

P(X=1) = e-2.4(2.41/1!) = 0.217723087

P(X=2) = e-2.4(2.42/2!) = 0.261267705

0.0907+0.2177+0.2612 = 0.5697 (56.97%)

(ii) At least 1, Therefore we need to examine

*P(X=0)*. Then 1-P(X=0)

P(X=0) = e-2.4(2.40/0!) = 0.090717953

1-0.090717953 = 0.90929 (~90.93%)

(iii) At least 1, Therefore we need to examine

*P(X=10)*. Over 7 years

P(X=0) = e-2.4(2.40/0!) = 0.090717953

P(X=1) = e-2.4(2.41/1!) = 0.217723087

P(X=2) = e-2.4(2.42/2!) = 0.261267705

P(X=3) = e-2.4(2.43/3!) = 0.209014164

P(X=4) = e-2.4(2.44/4!) = 0.125408498

P(X=5) = e-2.4(2.45/5!) = 0.060196079

P(X=6) = e-2.4(2.46/6!) = 0.024078431

P(X=7) = e-2.4(2.47/7!) = 0.008255462

P(X=8) = e-2.4(2.48/8!) = 0.002476638

P(X=9) = e-2.4(2.49/9!) = 0.000660436

P(X=10) = e-2.4(2.410/10!) = 0.000158504

**Part (iii)**is where I run into an issue. The question states it must be exactly 10 tornadoes in the space of 7 years. Meaning there could be any combination of tornadoes in the years. For example; the first year could have all 10 tornadoes, with the following years not having none. Or the first year could have none, the second year could have 3, then the remaining 6 tornadoes in the following years.

How I was going to attempt this final part was by summing these values up. This would give the maximum probability per year, then multiply this by 7 (as 7 years) I doubt this is correct.

Where does the 7 years come into the formula? and is it correct to assume that calculating P(X= 1 to 10) is relevant to the question?