In any given year, the probability of an earthquake greater than Magnitude $6$ occurring in the Garhwal Himalayas is $0.04$. The average time between successive occurrences of such earthquakes is ____ years.
My attempt:
Somewhere, answer is given $25$ years.
This means $4$ out of $100$ years will face an earthquake greater than Magnitude 6.So average time between successive earthquakes will be $\frac{100}{4}=25$ years.
I'm not getting how I apply $\frac{1}{0.04}=25$ with given probability and average time.
Can you explain in formal way? Please.
Best Answer
Let $X$ be the number of years to the next earthquake.
Then $X$ is a geometric random variable and the
probability that the next earthquake happens in $k$ years is
$P(X = k) = (1-p)^k \cdot p$
where $p = 0.04$.
The expected value of $X$ is
$E(X) = 1/p = 1/0.04 = 25 \:\text{years.}$
And this can also be interpreted as if you waited a long period of time the average time between earthquakes would be $25$ years.
(or you can think of it that the distribution of the length of the gaps between earthquakes will follow a geometric distribution)