## Trivia

Larry Barrett

The complete set of cases (using bold to indicate the event and italics to indicate the 'not x' event*) is *A**B**C*, **A***B**C*, **B***A**C*, **C***A**B*, **A****B***C*, **A****C***B*, **B****C***A*, **A****B****C**.

When observing for 2 hours, **A** is certain to erupt so the probability of all events which include *A* is 0, which eliminates all but four: **A***B**C*, **A****B***C*, **A****C***B*, **A****B****C**.

Alex has already shown us the probability for the **A****C***B* event occurring in a 2 hour period: 1*1/2*1/3= 1/6, and the probability that **A** occurs first is 1/2*1/6.

The probability of **A***B**C* occurring is 1*1/2*2/3 = 2/6, and the probability that **A** occurs first is 1*2/6.

The probability of **A****B***C* occurring is 1*1/2*2/3 = 2/6. and the probability that **A** occurs first is 1/2*2/6.

The probability of **A****B****C** occurring is 1*1/2*1/3 = 1/6, and the probability that **A** occurs first is 1/3*1/6.

So the probability that **A** occurs first in the 2 hour observation period is

1/2*1/6 + 1*2/6 + 1/2*2/6 + 1/3*1/6 = 3/36 + 12/36 + 6/36 + 2/36 = 23/36.

Similar calculations for the prob of **B** occurring first = 8/36 and for **C** occurring first = 5/36. The sum of all three probabilities is 23/36+8/36+5/36= 36/36= 1.

*My keyboard is configured so that the key which contains the 'not' symbol is used to toggle between the english and thai alphabets, so I needed another way to indicate the 'not x' event.