# Week 1, 9/2/08, Probability of having a good day

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Zeus, Athena, and Poseidon visit Mount Olympus exactly once every day. They each arrive at a random time between 8:00 AM and 5:00 PM, and stay for 2 hours. If two of the three gods meet at Mount Olympus on a given day, then the next day will be a good day on Earth.

Here, we are ignoring the third god, so only Zeus and Athena visit Mount Olympus...

Either god can arrive at any moment between 8 a.m. and 5 p.m., or rather, on the interval ${\displaystyle [0,9]}$. The interval can be split into 3 regions:

${\displaystyle 0\leq x<2}$,

${\displaystyle 2\leq y\leq 7}$

${\displaystyle 7

Looking at the ends of each region we can see the probability that the two gods overlap given that one has arrived at these points. So,

${\displaystyle P(B|0)={\frac {2}{9}}}$

${\displaystyle P(B|2)={\frac {4}{9}}}$

${\displaystyle P(B|7)={\frac {4}{9}}}$

${\displaystyle P(B|9)={\frac {2}{9}}}$.

The graph looks like:

Integrating and dividing by 9, the probability that two gods will meet is ${\displaystyle {\frac {32}{81}}}$