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Blue Moon (Posted on 2007-06-07) Difficulty: 3 of 5
The most common current definition of a blue moon is the second full moon of a calendar month.

Given that the full moon occurs every 29.5 days what is the long term probability that when a blue moon occurs it is in a specific month? Use the regular (every 4th year) leap year formula for February.

Consider a world/calendar with a lunar cycle of exactly 29 days and whose months are randomly assigned 30 or 31 days (with equal probability and no yearly repetition.) What is the long term probability that a blue moon falls on a 30 day month?

See The Solution Submitted by Jer    
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Solution solution Comment 1 of 1
Part 1:

Though actual lunations can vary from about 29 days, 6.5 hours to about 29 days, 20 hours, averaging 29.53 days, we'll use the 29.5 for the purpose of this puzzle.

29.5 days is 59 half-days, while four years is 365.25*4*2=2922 half-days. These two numbers are relatively prime, so eventually all relative phases (in increments of 1/2 day) should occur. There is zero probability that full moon would occur at exactly the beginning of a year, so we might as well consider, say, a 1/4 day offset, so that a full moon will occur halfway into a particular half-day. It could be any offset greater than 0 and less than 1/2.

In a 30-day month there is an excess of 1/2 day. Only if a full moon occurs within that half day will it be blue. That has a probability of 1/59, just as any half day has of containing a full moon.

A 31-day month has an excess of 3/2 days, so it has a 3/59 probability of containing a blue moon.

Neither 28- nor 29-day months can have a blue moon.

The question seeks to know what the probability distribution is, over the months, of a given blue moon being in different months. The relative probabilities are:

January   3
February  0
March     3
April     1
May       3
June      1
July      3
August    3
September 1
October   3
November  1
December  3

The column adds up to 25. So the probability that a given blue moon will be in January is 3/25 (and likewise for the other months with 3 next to their names), and the probability that a given blue moon will occur in April is 1/25 (and likewise for any month with a 1 next to its name).

Part 2:

A 30-day month has a 1/29 chance that its 30th day (and therefore its first day also) has a full moon. A 31-day month has a 2/29 probability.  As there would be an equal number of 30- and 31-day months, a given blue moon would have a 1/3 probability of finding itself in a 30-day month, not saying which one, and a 2/3 probability of finding itself in a 31-day month.

  Posted by Charlie on 2007-06-07 16:56:31
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