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This is question 3.16 from the book "Quant Job Interview Questions and Answers" by Mark Joshi et al. The solution given in the book states that for the fourth business day of the month to be Thursday, the first day of the month should be either Saturday, Sunday, or Monday. Therefore the answer is 3/7.

Although the authors haven't mentioned this, they seem to assume that the first day of the month is random, which I feel is not a correct assumption. A year consists of 365 days. 365=1 modulo 7. Therefore, every year the day shifts by one, except for leap years where the following year is shifted by two. For example, May 1st 2006 was a Monday, May 1st 2007 was a Tuesday, May 1st 2008 was a Wednesday, but May 1st 2009 was a Friday as 2008 was a leap year, and so on. This looks like a part of a repeating cycle. We should track one full cycle and then count the number of months in that cycle with Saturday/Sunday/Monday as the first day and divide that by the total number of months in the cycle to get the answer (We need to consider nuances such as years divisible by 100 not being leap whereas years divisible by 400 being considered as leap years).

Is there something wrong with my reasoning? Or is 3/7 indeed the right answer? If so, how can we assume that the first day of a given month is random? Is it a valid assumption?

J. W. Tanner
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bobby202
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    Your alternate perspective is perhaps more appropriate if a specific range of months is defined (i.e. the probability of the 4th business day being thursday, when a month is chosen at random from the interval jan-1900 thru dec-2099). – user2661923 Nov 24 '23 at 02:05
  • Regardless of any specific range, I feel this is cyclical i.e. we should come back to the same date-to-day configuration for some year in the future. For example, if January 1st 2023 was a Sunday, then there must be some other year in the future that's exactly before a leap year, has 365 days, and whose first day is Sunday. This kind of makes the assumption of random first day somewhat unreliable. – bobby202 Nov 24 '23 at 03:25
  • The punchline is that your intuition is correct that it might be skewed and it is a priori not a correct assumption to think that each day of the week is equally likely to start the month. To answer your specific question, you will want to first clarify what the question is... if talking about a specific range then that would be fine. If talking about time immemorial... (ignoring the fact that "work weeks" don't really exist after our robot overlords enslave us in the future) then that is equivalent to picking from a day within a 400 year cycle at random... – JMoravitz Nov 24 '23 at 03:29
  • You'll probably want to write a python script or some other program in whatever your preferred language is at that point and just brute force the answer. That being said... the true value will surely be very close to the $\frac{3}{7}$ answer that the naive simplification would give, and some times a dirty approach with rough approximations is perfectly acceptable. – JMoravitz Nov 24 '23 at 03:31
  • This answer was about the day of the week used for 13'ths of the month. Noting that the 15th is on the same day of the week as the 1st of the months, we can shift the table found there by two days, giving the exact answer as $\dfrac{685+685+687}{4800}=\dfrac{2057}{4800}= 0.428541\overline{6} < 0.\overline{428571}=\dfrac{3}{7}$ – JMoravitz Nov 24 '23 at 03:39
  • @JMoravitz Thank you so much. The Friday the 13th answer made a lot of sense. So there is a slight skew, but they're all hovering around 685, so I guess they can be considered to be equally likely for all practical purposes. Thanks a lot!!! – bobby202 Nov 24 '23 at 07:03

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