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 
year would start with a Sunday?

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/