Probability that a geyser erupts > Lets say you have a geyser that has a 2/3 probability of erupting in a 50 minute interval? What is the probability that it will erupt in a 20 minute interval? The way I tried to solve it that a 20 minute interval is 2/5 of a 50 minute interval, so the probability is 2/3 * 2/5 = 4/15, but apparently this is worng. Where did I go wrong and what is the right method to solving it?

Let us assume the event (random variable) of geyser erupting within 50 minutes follows an exponential distribution

In other words P(X<=50minutes)= $1-e^{-\lambda (50)}$ = 2/3

Find lamda from this, The lamda is somewhere around 0.021972.

Now find P(X<=20 minutes) = $1-e^{-\lambda * (20)}$ = 0.35561.

That will be your answer.

Alternate Slick Answer:

Divide the 50 minute interval into five 10-minute intervals. Let Q be the probability that the geyser does not erupt in each of the 10 minute intervals. Thus, $Q^5 = \frac{1}{3}$ and $Q \approx 0.8027$. For 20 minutes it is just $1−Q^2$ or 35.56%.

Thanks

Satish