Years That Start on SundayDate: 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/ |
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