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Alright math wizards, this is probably a fairly simple one... I'm interested in knowing what the probability is that a person has their $13$th birthday on Friday the $13$th (in any month). I'm especially interested in the formula used to solve this.

Here's my best guess. There is (roughly) a $1$ in $30$ chance that a person is born on the $13$th day of any given month. There is a $1$ in $7$ chance that a person's $13$th birthday occurs on a Friday. Multiplying these two probabilities $(7 \times 30)$ yields roughly a $1$ in $210$ chance that a person's $13$th birthday occurrs on Friday the $13$th.

TopoSet32
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Kabron
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    What may we assume about probability of date of birth? The Gregorian calendar repeats once every 146097 days. Should we assume that the person is equally likely to have been born on any one of those 146097 days? Or simpler than that, equally likely to have been born on one of the 365 days of a specific year, say year 2001. In reality, certain world events (such as wars) or specific seasons (such as winter) will cause people to become more sexually active causing the real probabilities to be skewed. – JMoravitz Oct 25 '15 at 00:08
  • You may be interested to read this post about the probability christmas is on a weekend, as well as this link to scienceworld.wolfram about friday the thirteenth's frequency in our calendar. – JMoravitz Oct 25 '15 at 00:09
  • I am not interested in a sophisticated probability equation, just a simple probability equation that doesn't control for things like wars, seasons, leap day, and so on. Assume equal probability that the person is born on any given day of the calendar. – Kabron Oct 25 '15 at 00:13

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$\frac{1}{210}\approx 0.00476190$

There is a slight error in your calculation due to your assuming "thirty days on average" for a month, where in fact it is closer to $\frac{365.24}{12}\approx 30.437$ days per month on average.

Using this to correct your solution, we have have a seemingly more accurate approximation as $\frac{1}{30.437\cdot 7} \approx 0.0046935$

The solution that you suggested is close to accurate. However, due to the nature of our calendar, friday the thirteenth is slightly more probable than one might expect. Read this page for more. The thirteenth of the month is more likely to be a friday than any other day of the week (by a fraction of a percentage, but still).

Our calendar system repeats once every 400 years (or 146097 days). Out of the 146097 days in the gregorian calendar, exactly 688 of them are friday the thirteenths. We may rephrase the question from "thirteenth birthday is friday the thirteenth" to "is born on friday the thirteenth" (ignoring chance of death as a child).

Assuming each day of our calendar is equally likely then, the probability should be $\frac{688}{146097}\approx 0.0047092$.

JMoravitz
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  • So by rephrasing the question are you suggesting the probability of being born on Friday the 13th is the same probability as a 13th birthday occurring on Friday the 13th? – Kabron Oct 26 '15 at 20:12
  • Yes. In particular, it is simply the probability that a randomly selected day is a Friday the thirteenth and that day can have some significance to the person (be it their birth, their thirteenth anniversary of their birth, etc) – JMoravitz Oct 26 '15 at 20:59
  • In reality these probabilities are slightly different - if someone's 13th birthday is on Friday the 13th, they'll have been born on Tuesday or Wednesday the 13th (depending on where their birth is relative to the leap year cycle) and there are slightly more births on Tues/Wed than on Fri (see e.g. https://blogs.scientificamerican.com/sa-visual/why-are-so-many-babies-born-around-8-00-a-m/) - but of course this is irrelevant to the simple probabilistic model that the OP said they wanted used.) – Michael Lugo Jun 27 '17 at 20:01
  • @michaellugo are you suggesting that there are somehow more tuesdays the thirteenth or Wednesdays the thirteenth where thirteen years later is a Friday the thirteenth than there are Friday the thirteenths themselves? Do not confuse the number of Tuesday the thirteenths with Tuesday the thirteenths who have a Friday the thirteenth thirteen years later. – JMoravitz Jun 27 '17 at 20:32