## Five Mondays

Puzzle. What is the probability of getting five Mondays in a 31-days month?

This is easy if we assume that the day of the week for the first day of the month is chosen at random. But we should know better. What is the actual probability? Bonus question: for which day of the week the probability of having this day five times in a 31-days month is the highest?

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1. #### David Reynolds:

p(5 Mondays in a month) = 385/4800 = 0.08020833333+ Most common for August or September.

I’m puzzled by your bonus question, since I calculate a three way tie:

p(5 Sundays in a month) = 387/4800 = 0.080625. Most common for January, May, or July.
p(5 Tuesdays in a month) = 387/4800 = 0.080625. Most common for January, May, or December.
p(5 Fridays in a month) = 387/4800 = 0.080625. Most common for January, May, or July.

2. #### David Reynolds:

My last post was wrong in a couple of ways. I was only considering months that started with the day in question. And I had February as the wrong month so my leap calculations were off.
I now calculate the odd of five Mondays in a month to be 1671/4800 = 0.348125
And I get the same answer for any day of the week.

3. #### Puzzled:

Can we rely on the Gregorian calendar for this puzzle? If not, which one is to be used?

4. #### J:

It’s trivially 1, but it actually depends.

5. #### David Reynolds:

Another correction.
Any month of the year is capable of having 5 Mondays. Even February, but that doesn’t happen very often. If you consider all months, then my calculated probability of 1671/4800 = 0.348125 is correct.
But the problem stated that only 31 day months should be included.
Therefore, the probability of 5 Mondays in a 31 day month is 1198/2800 = 0.427857+.
The probability of 5 Thursdays in a 31 day month is 1202/2800 = 0.4292857+ which is the answer to the bonus question.
172 each of January, March, May, October, and December and 171 of July and August in 400 years.

6. #### tanyakh:

Yes, I meant Gregorian calendar, but I didn’t mention it, because I didn’t want to give a hint. My calculations are the same as the last comment from David.

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