The Probability that the Last Day of a Randomly Chosen Century is a Thursday
In this article, we will explore the mathematical probability that the last day of a randomly chosen century is a Thursday. We will break down the analysis into several steps, covering the length of a century, counting days, and calculating the probability.
Step 1: Understanding the Length of a Century
A century consists of 100 years. Most years have 365 days, but leap years add an extra day every 4 years, typically. However, there are exceptions for years divisible by 100, which are not leap years unless also divisible by 400. This rule is part of the Gregorian calendar.
Step 2: Counting Days in a Century
Let's delve into the detailed calculations for a typical 100-year period:
Normal Years
In a 100-year period, there are usually 75 normal years (365 days) and 25 leap years (366 days). This is because the leap year system ensures that every 4 years an additional day is added.
Step 3: Calculating the Total Days in a Century
Let's calculate the total number of days for both scenarios:
Century with 25 Leap Years
Total days 75 times 365 25 times 366 27375 9150 36525 days.
Century with 24 Leap Years
Total days 76 times 365 24 times 366 27390 8784 36504 days.
Step 4: Distribution of the Last Day of the Century
The last day of a century, specifically December 31, can be analyzed by considering the starting day of the century. To determine the day of the week, we take the total number of days modulo 7.
Century with 25 Leap Years
36525 mod 7 1, which means December 31 is 1 day after the starting day.
Century with 24 Leap Years
36504 mod 7 4, which means December 31 is 4 days after the starting day.
Step 5: Probability Calculation
The probability that the last day of a randomly chosen century is a Thursday requires a uniform distribution over the 7 days of the week. Each starting day of the century is equally likely to be any day of the week:
- If a century starts on a Sunday after 25 leap years, December 31 is a Monday.
- If a century starts on a Sunday after 24 leap years, December 31 is a Thursday.
- Since every starting day leads to each of the 7 days of the week being equally likely, the probability is:
[text{Probability} frac{1}{7}]
This results in a probability of approximately 0.142857 or 14.29%.
Conclusion
The probability that the last day of a randomly chosen century is a Thursday is 14.29%, given the uniform distribution across the seven days of the week.