Probability of rare events with some time parameter

Question

While trying to $solve$ a simple problem involving probability theory, I run in some difficulties. The question is simple:

A type of earthquake is verified to occur in the long run once in $50$ years with probability of $75 \%.$ What's the probabilty that it will occur once in $10$ years ?

I first thought to use the Bayesian statistics on conditional probabilities, but I could not convince myself. I could not formulate apropriately the given data and adapt it to the required data for the Bayesien formula.

I then $thought$ of the Poisson distribution, which might eventually model the occurrence of rare events with a time parameter. I did not find an indication though that such an earthquake will follow the Poisson distribution.

I will appreciate your help.

Answer 1

if the rate of poisson distribution is lambda
probability of occurring of earthquake in time t is P(X >= 1)
= 1- P(X = 0)
= 1 - e^(-lambda * t)

now it is given for t = 50

1 - e^(-lambda * 50) = 0.75
e^(-lambda * 50) = 0.25
find lambda from this equation.

and you need to find the probability for t = 10

hence required probability is

1 - e^(-lambda * 10)
just substitute the value of lambda from above.

Thanks