I was born on April the 13th 1990. The day was also Good Friday. I'm wanting to know what the probability/chances of being born on a Good Friday that is also a Black Friday is/are to have an understanding of how rare such a day of birth is.
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1$\begingroup$ First, "friday the thirteenth" is not what we call "black friday". That has a very different meaning. Second... "what is the probability"... you need to be clear as to what assumptions are being made such as assuming a person is equally likely to be born on any day of the year and assuming the specific year they are born in is also equally likely to be any year over a sufficiently large number of millennia so as to make the limiting probability converge... $\endgroup$– JMoravitzCommented Apr 4 at 15:57
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1$\begingroup$ Third... coincidences happen all the time. If you weren't commenting on how surprising this specific situation is, there would still have been some other situation just as surprising you could have been commenting on. Finally... actually going through the effort to find the exact probability is going to be incredibly tedious. Approximations are going to make this much easier... such as noting that Easter falls on a day from Mar22 to Apr25 with close to equal proportions, and a 13th of the month being a particular weekday with close to equal proportions. $\endgroup$– JMoravitzCommented Apr 4 at 16:01
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2$\begingroup$ Good Friday is on a Friday the 13th if and only if Easter is on April 15. This has occurred in 1900, 1906, 1979, 1990, and 2001; and will occur again in 2063, 2074, 2085, and 2096. $\endgroup$– Geoffrey TrangCommented Apr 4 at 16:23
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2$\begingroup$ @Henry I was well aware and I explicitly stated I was making approximations, which is something the OP could have been expected to have been able to do to arrive at an answer. I also explicitly gave the link with full knowledge of the contents and that it does give a more accurate number and alluded to the fact the link could have been used for a more accurate approximation. You would notice I also did not use $365.2425$ in my approximations. $\endgroup$– JMoravitzCommented Apr 4 at 16:54
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2$\begingroup$ Related question: How to reasonably estimate the probability of your father being exactly 12222 days older than you? $\endgroup$– AmirCommented Apr 4 at 22:09
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2 Answers
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From
The Easter date formula dictates that is always the first Sunday after the first full moon that falls on or after March 21.
Easter is a moveable feast, which means that it falls on a different date each year. Easter always falls on a Sunday, but Easter Sunday can be as early as March 22 and as late as April 25.
This means, as Good Friday is two days before Easter it can only fall on Friday April 13th to satisfy your conditions. Then Easter is on April 15th, so the full moon must fall between April 7th and April 14th that year and April 15th must be a Sunday.
This works out at $\frac{7}{35}*\frac{1}{7}$ =$\frac{1}{35}$ (See correction below)
Day of the week for any date changes pretty regularly each year as does the date of the full moon for any month. So if you assume that Easter is equally likely to fall on any date between March 22nd and April 25th in any particular year. That’s 35 days. The probability that it falls on April 15th is then 1:35. Or approx 0.0286.
However:
A historical survey shows that between 1600 and 2100 there will be a total of 18 times that this has occurred i.e probability is 18/500 = 0.036
The mean number of times this has occurred each century is 0.036, with a SD of 0.0652. This means that the calculated value of 0.0286 lies outside of one standard deviation from the norm.
EDIT: Correction
As @Henry points out the likelihood of Easter falling on the first 6 days or last 6 days is lower than on other days. Also, the ecclesiastic calendar fudges times of the full moon, so that Easter never falls on April 26th.
Besides this, I made a mistake in my original calculation; the probability of the full moon falling in the 7 days before April 15th is in fact 7/(days in synodic month) and not 1/35! So the probability of Easter falling on April 15th is 0.03386 which is in close agreement with the average over those 5 centuries.
If we assume that the probability of being born on any particular day of the year is equally likely, then the probability of being born of Good Friday, the 13th of April is: $\frac{1}{days in lunar month}*\frac{1}{days in tropical year} = .0000927$ or about 1:10000.
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Since Easter is always the Sunday two days after Good Friday, this equation is equivalent to asking “How often does Easter fall on April 15?”
By brute-force calculation of the date of Easter within 200 years of the present (i.e., 1824 to 2224), it happens 16 times (1827, 1838, 1900, 1906, 1979, 1990, 2001, 2063, 2074, 2085, 2096, 2131, 2142, 2153, 2210, or 2221) in those 401 years, or in 3.99% of years.
If you assume that birth dates are uniformly-distributed within the 401-year (146462-day) interval, then the probability of being born on Good Friday the 13th works out to $\frac{16}{146462}$, or approximately $0.000109$.
Edit: If you consider the entire 5,700,000-year cycle of Gregorian Easter dates, Easter falls on April 15 in 192850 of those years, or $\frac{203}{6000}$ of the time. Thus, the long-term probability of being born on a Good Friday the 13th works out to approximately $0.0000926$.
