Lela's Birthday is a "Lela Birthday"

I would like to propose a mathematical concept I call a "Lela Birthday."

My friend Lela was born on 4/9/49. She just celebrated birthday 49 on 4/9/98.

It is a rare event when someone whose birth date had the structure M/D/MD celebrates her MDth birthday on M/D/2xMD, where M and D are digits. How rare? I will try to calculate the percentage of birthdays that are "Lela birthdays".

First, what percent of the population is born on a date with the structure M/D/MD where M and D are non-zero digits? In any century, 19 of the 100 years have 0 in them, so 81% have the structure MD. In any year, 9 of the 12 months have the single digit structure M. In any month, 9 of the roughly 30 days have the single digit structure D. Of these people, about 1 out of 365 has their month/day M/D corresponding to their year MD. If we assume an average life expectancy of 80 years, then people on average do not live to celebrate the 17 birthdays from 81 through 99 (90 does not count), reducing the 81% to 64%. So the percentage of celebrated Lela birthdays is 64/100 x 9/12 x 9/30 x 1/365 = .04% or 1 birthday party in 2,500.

Now Lela turned 49 in '98, where 98 = 2 x 49. It is always true that a candidate for a Lela birthday turns MD years old MD years after the year 'MD. So, in most even numbered years from 1922 to 1998 (except 1940, 1960, 1980), Lela birthdays took place where the year (e.g., '98) was twice the person's age (e.g., 49). This will no longer be true for people born in 1951 to 1999 since they will celebrate in 2002 to 2098. Of course, these special Lela birthdays will resume in 2022 for people born on 1/1/2011 and later. In general, then only 35/81 of Lela birthdays as calculated above have this special double-date feature, or .017% = about 1 birthday party in 6,000. The Lela birthday yesterday will be the last such very special birthday until 2011. If someone went to a birthday party every week of their life they would be lucky to attend one Lela birthday -- but now they would have to wait about 24 years for the chance.

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