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### Leap Day, and the Probability of being born thereon

On Leap Day 2008 (February 29 2008), Delaware's News Journal ran a front page article called, "Leap Day babies have an emperor to thank for their 'youth'". It included this passage:

Still, Iezzi-Farkas and Roberts [both Leap Day babies] like their rare birthdays. "I don't know what the odds are," he said, "but I'm sure it's a pretty small chance."

Tom Ilvento, who chairs the University of Delaware's statistics program, says the odds are 1 in 1,464 or 1 in 1,461, not counting the factor of century years being skipped to make up for odd minutes of the solar year. Some say that cuts the odds to about 1 in 1,500.

Ilvento calculates that a person is more than 400 times more likely to be born on Leap Day than to be hit by lightning (about 1 in 600,000) and about 10,000 times more likely to have that birthday than to win Powerball. Of 560,458 Delawareans registered to vote, 418 list Feb. 29 as their birthday - about 1 in 1,340. Nationwide, there are said to be about 200,000 Leap Babies, and more than 4 million worldwide.

Isn't that bizarre, two different probabilities? A sidebar discussed the probability calculations in more detail:

Tom Ilvento, who chairs University of Delaware's statistics program, says he and statisticians he consulted know two calculations used to figure the odds of being born on Leap Day.

The first starts with a 1 in 4 probability of being born in a Leap Year, he said, multiplied by the 1 in 366 chance of birth on any day of such a year.

"That gives you a combined probability of 1 out of 1,464," he said.

The easier calculation uses just the days in four years (365 times four plus one for Leap Year) for a probability of 1 in 1,461.

The odds are off a smidgen; he admits, due to rare Leap Year adjustments.

The solar year runs about 11 minutes shy of exactly 365 1/4 days - at 365.24219 - so Leap Year is skipped three times every four centuries to make up the minutes, a tweak made by Pope Gregory XIII in 1582. In even centuries, only those divisible by 400 get Feb.29 added. That made 2000 the first century year since 1600 to be a Leap Year. The next is 2400.

But Ilvento said century adjustments cause only a "minuscule" reduction in the probability of being born on Leap Day.

I thought that called for comment, so I sent a letter to the editor. As usual, newspapers don't know from interesting, and the letter was tossed. So now I take my case before the whole world . . .

Dear [Delaware] News Journal,

In your article on Leap Days, two different probabilities were calculated for being born on February 29. That should set off alarms in an educated mind. One of the beauties of mathematics is that you get the same answer no matter which route you take solving a problem. We're not talking psychology or astrology here - or even that "fuzzy math" stuff not being learned by modern schoolkids.

For the Leap Day problem, one solution described was to simply add up the number of days in a four-year span. (4x365) + 1 = 1461 days, so the probability of being born on Leap Day is 1 out of 1461.

The other approach was to multiply the probability of being born in a Leap Year (1 out of 4) by the probability of being born on Leap Day (1 out of 366). (1/4) x (1/366) = 1/1464, and we say the probability is "1 out of 1464."

Why the discrepancy? [In case a quick reading of the sidebar led one to think it has to do with the rare Leap Year adjustment, it does not. The adjustment did not figure into either of the approaches.]

The error is the "1 out of 4" probability for being born in a Leap Year. Since it's a longer year, it's a slightly bigger "target" to hit, and the probability is proportionally greater. To be specific, the Leap Year comprises 366 days out of the 1461 days total, as opposed to 365 out of 1461 for a normal year.

So to finish the problem off, multiply (366/1461) x (1/366) to get the same probability as in the other method, 1 out of 1461.

And all is well again in the math world.

A lot of effort to correct a "little" math error, writing a letter to the editor and then putting up a web page? Mebbe. But I can't help thinking that there is something really rotten in math education nowadays if even such a piddly, non-life-threatening mistake like that can sneak by so many people, including professional statisticians - or even a newspaper staff.

You don't get two different answers to the same math problem, people.

***

That was a good zinger to stop on, but while we're here, let's polish off the bigger problem of randomly picking a Leap Day, incorporating the Leap Year adjustments.

As explained in the sidebar, only century years divisible by 400 get Feb 29 added. Perhaps it's easier to think in terms of the century years NOT divisible by 400 as being the odd men out - the years that you would expect to have a Leap Day but do not get one.

So, over the course of 400 years - think of 1601 through 2000, if you'd like - you would have 100 leap years if there were no adjustment, but the adjustment knocks out 3 of them, making 97 leap years and 303 regular years. That makes for a total of

(97 x 366) + (303 x 365) = 146097 days
in a 400-year span.

Or, you might have saved a step by thinking, there are 400 years of 365 days each, plus 97 Leap Days:

(400 x 365) + 97 = 146097 days.

It's not magic, just your 5th-grade commutative, associative, and distributive properties for addition and multiplication at work.

Then on to the final answer with the understanding that 97 out of those 146097 days represent "hits", or "successes", probabilistically speaking. In decimal form, the probability of randomly picking a Leap Day is:

P(Leap Day) = 97/146097 = .0006639

Putting the probability in the "one in" form that we are more comfortable with,

P(Leap Day) = 97/146097 = 1/1506 = "one in 1,506"

(Both the .0006639 and 1506 numbers are rounded off to four places.)

Remember that the probability calculated without the Leap Year adjustment was somewhat better (one in 1,461). The adjustment takes away Leap Days.

After doing the math, I found myself looking for an intuitive way to view the difference between the probabilities calculated with and without the adjustment. Ilvento said the Leap Year adjustment would reduce the probability only "a smidgen" and I'll admit I was surprised to see a difference as large as we do. Here it's easiest to work with decimals. The probability calculated without the adjustment is 1/1491 = .0006845; the probability calculated with the adjustment is 97/146097 = .0006639

Relative to the first probability, the second probability is only

.0006639/.0006845 = 6639/6845 = .97 = 97%
as large. (Again, .97 is rounded off.)

Does that feel right?

You bet. Ignoring the Leap Year adjustment, our 400-year span has 100 Leap Days; working in the Leap Year adjustment, it has only 97 Leap Days. So a probability 97% as big is right on the button, intuitively. (It's not exactly 97% due to the tiny difference in the denominators of our probabilities. The probability is 100/146100 without the adjustment; and 97/146097 with the adjustment. The denominators are in agreement to four places, and the effect of their relative difference is tiny compared to the that of the numerators.)

While we're here, let's calculate the length of a day that fits perfectly with the Leap Year plan implemented by Pope Gregory XIII. The Gregorian Calendar gives us 146097 days per 400 years, which works out to 365.2425 days/year, exactly. Note how closely that agrees to the 365.24219 figure given for the length of the solar year in the sidebar. The difference is .00031 days, or 26.784 seconds. I've always enjoyed this paragraph in my Encyclopedia Americana under the entry for Calendar:

There is a remaining inexactness in that the Gregorian year is equal to an average of 365.2425 days and is therefore longer than the solar year by 0.0003 day per year. The excess amounts to 3 days every 10,000 years. But other factors come into play over such long periods. The solar year has been taken here as being exactly 365.2422 days long, but in reality it varies in length (although very slowly). The law governing this variation is imperfectly known. In addition, the rotation of the earth on its axis is subject to variations, some of which cannot be predicted. It would be futile to anticipate the calendar corrections that might be necessary in 3,000 to 4,000 years, when far greater disparities may intervene in the meantime. The Gregorian calendar therefore leaves nothing to be desired in its precision.

Gregory XIII was the man.

***

Finally, please help me out with a project I am working on. I am compiling a list of interesting events that occurred on September 11 1752. If you would send me one from your home town I would be most appreciative and will add it to a section in this web page in the future.

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