# What is the probability of a person born in leap year

1. Jul 1, 2007

### j_mohana6

what is the probability of a person born in leap year

2. Jul 1, 2007

### mathman

Slightly more than 1/4.

3. Jul 1, 2007

### j_mohana6

can u pllz explain me

4. Jul 1, 2007

### HallsofIvy

Staff Emeritus
I'm afraid I'm going to have to disagree with mathman. It is slightly less than 1/4. The answer is not exactly 1/4 because, in the Gregorian Calendar (as opposed to the Julian Calendar) every year which is divisible by 100 but not divisible by 400 is not a leap year. The year 1900, even though it is divisible by 4 was not a leap year, but the year 2000, which was divisible by 400 is not. If a leap year came every 4 years, then in 400 years, there would be 100 leap years. If any year divisible by 100 were not a leap year, that would reduce it to 100- 4= 96. Because the 400th year is divisible by 400, we put that back in: there are 97 leap years in 400 years. I think those are the only conditions. If so, the probability that an arbitrarily chosen year is a leap year is 97/400= 0.2425.

Last edited: Jul 1, 2007
5. Jul 2, 2007

### D H

Staff Emeritus
The problem is not that an arbitrarily chosen year is a leap year. It is that an arbitrarily chosen point in time is a leap year. While 97/400 years are leap years, a leap year has one extra day than a normal year. In a 400 year period, 97*366=35502 days will be in a leap year while 303*365=110595 days will be in a non-leap year. The probability that an arbitrarily chosen day occurs in a leap year is thus 35502/(35502+110595) = 0.243003.

6. Jul 2, 2007

### HallsofIvy

Staff Emeritus
Very good point.

7. Jul 2, 2007

### daveb

Are we talking about if you randomly pick a person off the street, what is the probability of them being born in a leap year? If that's what you want, then the only real way to be 100% sure of the probability is to tally each person on the planet with the year they're born. Then the probability of a person being born in a leap year is the number of people born in a leap year divided by the total number of people. So, yeah, it's probably easier to assume a randomly picked day is in a leap year or not.

8. Jul 2, 2007

### mathman

Since 2000 was a leap year, the probability that anyone alive today was born in a leap year is slightly over 1/4.

9. Jul 3, 2007

### haiha

We shoul calculate for all the time, not for any period of time. As I know, an average year's lengh for all time is 365.2425 days. An ordinary year consists of 365 days, a leap has 366.
Now let n the the number of ordinary years corresponding to one leap year. We have the following equation:

n*(365)+366 = (n+1)365.2425.

n is calculated as 3.1237
So the probability of being born in leap year is
P = 1/(n+1) = 1/(3.1237+1) = 0.2425 (strange?)

Correction:

P should be : 1*366/(n+1)365 = 0.2432

Last edited: Jul 3, 2007
10. Jul 3, 2007

### Kummer

Probability can be thought of
$$\frac{\mbox{ favorable outcomes }}{\mbox{ possible outcomes }}$$

In exactly 4 years there are: 366 favorable days and 1461 possible days (365+365+365+365+366)

To the probability is:
$$\frac{366}{1461} \approx .25051334$$

Note: Yes, I know leap years change every hundreds of years but I am not assuming that. I am doing it with an easy model.

11. Jul 6, 2007

### Krusty

What's the point of using a simple model when the full, exact model is no more complex?

12. Jul 8, 2007

### HallsofIvy

Staff Emeritus
In order to get an incorrect answer, of course!