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### Years That Start on Sunday

```
Date: 10/02/2000 at 17:36:16
From: Jason Lane
Subject: Probability that a year that starts with a Sunday

What is the probability that, if you were to pick a random year, that

I tried working it by thinking of Monday as an arbitrary starting
value. I knew that if a year starts on a Monday, it will end on a
Monday, and every four years (with the exception of every 100 years)
is a leap year, so that if the year begins on a Tuesday it will end on
a Wednesday, thus starting the next year on a Thursday. I calculated
that the pattern loops every 28 years, or 7 sets. Also, in these 28
years, there were 4 that started with a Sunday.

28 goes into 100 three times with a remainder of 16, so then you have
to calculate how many Sundays are in those 16 years, and I got 2. Then
on the 100th year I made both the starting and ending day of the year
the same because it's not a leap year.

I then looped it for 400 years, because every 400th year is a leap
year. And for the number of years starting with Sunday I got:

3(4)+ 2 + 3(4) + 3 + 3(4) + 3 + 3(4) + 2 = 58

The four sets alternate as 2, 3, 3, 2 in the number of Sundays in the
last 16 years. I finally got: 58/400 or 29/200.

I'm not sure if this is the answer, but I know that the procedure is
not entirely correct. I would appreciate if you would look over this
and tell me if I'm on the right track, and if not what I should do.

Thank you very much.
```

```
Date: 10/02/2000 at 18:10:13
From: Doctor Schwa
Subject: Re: Probability that a year that starts with a Sunday

You're doing really, really well. The question is a little bit hard to
answer; a random year out of which years? I think you can see from the
work you did that out of the next 28 years, exactly 4 start with
Sunday, so the probability is 4/28 = 1/7. Similarly if you look at the
next 28 years after that, you'd get the same answer. There is a little
confusion around those century years with the missing leap year in
them, but maybe you can figure out that if you look at a chunk of 2800
years that will have to even out too.

What I'm trying to say is that in the short run (200 or 400 years)
there may be a few extra Sundays here or there because of those weird
century things, but in the long run the century things are equally
likely to give you an extra Sunday as they are to take away a Sunday,
and overall everything evens out in the long run to make the
probability of Sunday exactly 1/7.

Is that convincing?

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 10/26/2000 at 15:52:23
From: Jason Lane
Subject: Probability that a year that starts with a Sunday

My teacher told me the answer to my earlier question was wrong...

I am spent on ideas. Can you help?

Thank you,
Jason
```

```
Date: 10/26/2000 at 23:06:13
From: Doctor Peterson
Subject: Re: Probability that a year that starts with a Sunday

Hi, Jason.

I saw your discussion with Dr. Schwa, and it sounded vaguely familiar.
When I looked in our Calendar FAQ page,

http://mathforum.org/dr.math/faq/faq.calendar.html

I found the claim that the 13th of a month falls more often on Friday
than on any other day. That implies that some configurations of the
year must happen more often than others. So your teacher is evidently
right, though I haven't yet run across anything on the Web that
discusses this aspect of the problem.

If you look at the explanation (in the link to Eric Weisstein), the
reason for this counterintuitive result is that the cycle of years is
cut short before you would expect, after only 400 years, which
contain a number of days divisible by 7, rather than the full 2800
years. Since 400 is not divisible by 7, there must be some types of
years that occur more often. It would take some calculating to
determine just how often each of the 14 types of years occurs, but it
could be done. You can try, and I might too.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 10/27/2000 at 09:05:56
From: Doctor Peterson
Subject: Re: Probability that a year that starts with a Sunday

Hi, Jason.

I worked on this a bit more, and wanted to share the details with you.
I hope you've done some more thinking too.

As I said, what we have to do is merely to count how many years begin
on Sunday during the 400-year period of the Gregorian cycle. The trick
for a mathematician is to do that with as little work as possible.

I started by making a list of the starting day of each year of the
cycle, in 28-year groups. Remember that

- the starting day advances by one day after a normal year, and by two
days after a leap year, since 364 days is a multiple of 7;

- 2000 is a leap year, but 2100, 2200, and 2300 are not, so the
pattern will change at those points;

- 2000 began on a Saturday (a memorable day for those of us in the
software industry who were on call for Y2K problems).

Writing 0 for Sunday, 1 for Monday, and so on, we get:

2000: 6123 4601 2456 0234 5012 3560 1345
2028: 6123 4601 2456 0234 5012 3560 1345
2056: 6123 4601 2456 0234 5012 3560 1345
2084: 6123 4601 2456 0234
2100: 5601 2456 0234 5012 3560 1345 6123
2128: 5601 2456 0234 5012 3560 1345 6123
2156: 5601 2456 0234 5012 3560 1345 6123
2184: 5601 2456 0234 5012
2200: 3456 0234 5012 3560 1345 6123 4601
2228: 3456 0234 5012 3560 1345 6123 4601
2256: 3456 0234 5012 3560 1345 6123 4601
2284: 3456 0234 5012 3560
2300: 1234 5012 3560 1345 6123 4601 2456
2328: 1234 5012 3560 1345 6123 4601 2456
2356: 1234 5012 3560 1345 6123 4601 2456
2384: 1234 5012 3560 1345
2400: 6...

As we were told, after 400 years we repeat the cycle. And since 400 is
not a multiple of 7, we know we CAN'T have the same number of years
starting on each day.

Now we can count the years in each group that start on each day:

0   1   2   3   4   5   6
Sun Mon Tue Wed Thu Fri Sat
2000:  4   4   4   4   4   4   4
2028:  4   4   4   4   4   4   4
2056:  4   4   4   4   4   4   4
2084:  2   2   3   2   3   1   3
2100:  4   4   4   4   4   4   4
2128:  4   4   4   4   4   4   4
2156:  4   4   4   4   4   4   4
2184:  3   2   3   1   2   3   2
2200:  4   4   4   4   4   4   4
2228:  4   4   4   4   4   4   4
2256:  4   4   4   4   4   4   4
2284:  3   1   2   3   2   3   2
2300:  4   4   4   4   4   4   4
2328:  4   4   4   4   4   4   4
2356:  4   4   4   4   4   4   4
2384:  2   3   2   3   2   3   1
--- --- --- --- --- --- ---
58  56  58  57  57  58  56 = 400

This confirms that years do not start on each day with the same
probability; Sunday, Tuesday, and Friday are most common, and Monday
and Saturday least. The answer to the original problem is that Sunday
occurs 58/400 = 29/200 = 14.5% of the time, which is a little more
than the expected 1/7 = 14.29%.

This illustrates an important fact about math: it can be easy to
"prove" something that isn't true, if we aren't careful. The mistake
Dr. Schwa made, which would have convinced me as well if I didn't have
reason to be suspicious, is to take familiar knowledge about the
behavior of modular arithmetic, which would apply here if leap years
were completely consistent, and assume that the "little confusion
around those century years" would not make a big difference.
Mathematicians, when they are not just answering little questions that
seem innocuous, never assume anything! Any change warrants careful
study.

Here's an extra project you might want to try: see if you can modify
my analysis a little and come up with the numbers given in our FAQ for
the frequency of Friday the 13th. You'll have to determine how many
Friday the 13ths there will be in each of the 14 kinds of years.

Also, check my numbers. If I've made any mistakes, or if your teacher
has a different way to approach this, please let me know, because I
suspect this will make its way into our FAQ, or at least the archives,
and we don't want mistakes there.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 10/27/2000 at 11:31:03
From: Jared Dervan
Subject: Re: Probability that a year that starts with a Sunday

I really appreciate the help that you have provided me. I have never
encountered something that could give me so much help using basic
reasoning.

I LOVE YOU GUYS.

Jason Lane
```

```
Date: 10/27/2000 at 12:10:45
From: Doctor Peterson
Subject: Re: Probability that a year that starts with a Sunday

Hi, Jason.

You're very welcome! You can probably tell that we enjoy the exercise
Dr. Math gives our brains, and interacting with people who are
interested in what we have to say makes it even better.

By the way, I didn't notice until after I sent the last response how
close you came to the right answer. You did a pretty good job, and
more "mathematically" than my cautious approach. Good work!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability