# Probability that a year contains 53 Mondays

The question:

Find a probability that a year chosen at random has 53 Mondays.

Now I know in a non-leap year, probability of getting 53 Mondays is $\frac{1}{7}$ and in a leap year, probability of getting 53 Mondays is $\frac{2}{7}$. Now knowing that leap year occurs after every 4 years, I felt the desired probability is $$\frac{1}{7}\times\frac{4}{5}+\frac{2}{7}\times\frac{1}{5}=\frac{4}{35}+\frac{2}{35}=\frac{6}{35}$$ However the solution given without any explanation is $\frac{5}{28}$. How is that? (I might be applying stupid logic with that $\frac{4}{5}$ and $\frac{1}{5}$).

• Under the now universally standard Gregorian calendar, one skips three leap years every four centuries, so the probability of a leap year is slightly less than $1/4$. The Gregorian calendar has been standard in predominantly Catholic countries since 1582 and in English-speaking countries since the 1750s. Some Slavic countries didn't adopt it until the 20th century. When the Julian calendar was adopted almost 21 centuries ago, the priests in charge of the calendar initially misunderstood the astronomer's instructions and had a leap year every three years${}\,\ldots\qquad{}$ – Michael Hardy Aug 30 '15 at 13:05
• $\ldots\,{}$because if A, B, C, D are consecutive years and A is a leap year, then D is the "fourth year", hence the next leap year. That seems to be the opposite of your misunderstanding. ${}\qquad{}$ – Michael Hardy Aug 30 '15 at 13:06
• When it comes to calculations then I advice you to ignore some (very interesting) comments here. – drhab Aug 30 '15 at 13:07
• @drhab : Another thing to ignore in calculations is this: "advice" is a noun and "advise" is a verb, and the final consonanat is unvoiced in the noun and voiced in the verb, just as with "belief" and "believe" and a number of other words in English. In some instances the difference is only in pronunciation and not in spelling; e.g. "I will use that idea." versus "I have a use for that idea." The word "use" is spelled the same way in both sentences, but pronounced differently. ${}\qquad{}$ – Michael Hardy Aug 30 '15 at 13:09
• @MichaelHardy In the meanwhile (spelled correctly, I hope) you are recognized as a serious English language enthusiast. Unfortunately Dutch and Latin are still missing in SE. My English is improving here day by day. – drhab Aug 30 '15 at 13:20

A leap year occurs every fourth year. That means your weights must be $\frac34$ and $\frac14$ resp. yielding $$\frac17 \cdot \frac34 + \frac27 \cdot \frac14 = \frac5{28}$$ The pattern of leap years is $$\Box\Box\Box\times\Box\Box\Box\times\Box\Box\Box\times\cdots$$
• @Mahesha999 Yes. Remember though, that the actualy probability will be slightly less because years divisible by $100$ but not by $400$ are "skipped" leap years. – AlexR Aug 30 '15 at 12:59