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?
 A: Your calculation would be fine if the geyser erupted exactly every $75$ minutes and the probability comes from the interval chosen.  The probability over an interval of length $t$ would be $\begin {cases} \frac t{75}& t \lt 75\\1 & t \ge 75 \end {cases}$  
You are probably expected to assume that the probability of eruption in a short interval of time $dt$ is $p\ dt$ and that the eruptions are independent.  Then the chance of no eruption over an interval of length $t$ is $e^{-pt}$.  Use the data you are given to find $p$, then evaluate $e^{-20p}$
A: 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
A: The answer is highly dependent on what your model is for geyser eruption.  Suppose you know it erupts exactly once per hour, and the unreliable person you asked says it erupted 15 minutes ago.  You are 2/3 confident that this is correct, but 1/3 confident it was actually 5 minutes ago.  This gives a 2/3 probability of eruption in the next 50 minutes, and a 0 probability of eruption in the next 20 minutes.
A: OK, I guess I'm late but will try nevertheless. All we know is that the probability of the event $P(S<75)=1$. This isn't enough to solve the problem, but given the situation (volcanic eruption) this follows some sort of exponential decay. So the mean $\frac{1}{\lambda}$ number of events in this period is $1$ if we take the time frame $=1$. The standardizing constant $\alpha$ is found from the equation 
$$
\int_{0}^{1} \alpha e^{-t}dt=1\\
\alpha=\frac{1}{1-e^{-1}}
$$
due to memorylessness property. Under these conditions the probability is 
$$
P(E)=\alpha \int_{0}^{\frac{20}{75}}e^{-t}dt
$$  
