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 . The interval can be split into 3 regions:

,

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,

.

The graph looks like:

Mtolympus2godgraph.png

Integrating and dividing by 9, the probability that two gods will meet is


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