The Curious Case of All-Even and All-Odd Number Sets: Rarer Than You Think

We often encounter sets of numbers in our daily 539 lives, from the digits on a clock to the prices in a grocery store. Sometimes, we might notice a peculiar characteristic: all the numbers in a particular set are either even or odd. This observation naturally leads to the question: how common are such “all-even” or “all-odd” number sets? While they certainly exist, a closer look reveals they are surprisingly less frequent than sets containing a mix of even and odd numbers.

To understand this, let’s first define what we mean by a “number set.” In this context, we’ll consider finite sets of integers. An “all-even” set is one where every number in the set is divisible by 2 (e.g., {2, 4, 6}, {-10, 0, 8}). Conversely, an “all-odd” set contains only numbers that leave a remainder of 1 when divided by 2 (e.g., {1, 3, 5}, {-7, 9, 11}).

Now, let’s consider a set of a specific size, say, a set containing $n$ distinct integers. For each number we choose to include in this set, there are two possibilities regarding its parity: it can be either even or odd. If we were to randomly select $n$ integers, the probability of any single number being even is approximately 1/2, and the probability of it being odd is also approximately 1/2.

To form an all-even set of size $n$, every one of the $n$ numbers we select must be even. Assuming the selection of each number’s parity is an independent event, the probability of this happening is $(1/2)\times (1/2)\times \cdots \times (1/2)$ (n times), which equals (1/2)n. Similarly, the probability of forming an all-odd set of size $n$ is also (1/2)n.

Therefore, the probability of a randomly chosen set of $n$ integers being either all-even or all-odd is the sum of these probabilities: (1/2)n+(1/2)n=2×(1/2)n=(1/2)n−1.

Let’s examine how this probability changes as the size of the set, $n$, increases:

• For a set of size $n=1$, the probability of it being all-even or all-odd is (1/2)1−1=(1/2)0=1. This makes sense because a set with only one number is trivially either even or odd.

• For a set of size $n=2$, the probability is (1/2)2−1=1/2. This means that out of all possible sets of two distinct integers, half of them will have both numbers even (e.g., {2, 4}), or both numbers odd (e.g., {1, 3}). The other half will have one even and one odd number (e.g., {2, 3}).

• For a set of size $n=3$, the probability drops to (1/2)3−1=1/4. Only one out of every four randomly chosen sets of three distinct integers will be either all-even (e.g., {2, 4, 6}) or all-odd (e.g., {1, 3, 5}). The remaining three-quarters will contain a mix of even and odd numbers.

As we continue to increase the size of the set, the probability of it being all-even or all-odd decreases exponentially.

• For $n=4$, the probability is (1/2)4−1=1/8.
• For $n=5$, the probability is (1/2)5−1=1/16.
• For $n=10$, the probability is (1/2)10−1=1/512.
• For $n=20$, the probability is (1/2)20−1=1/524288, which is incredibly small.

This demonstrates that as the number of elements in a set grows, the likelihood of all those elements sharing the same parity becomes increasingly rare. Sets containing a mix of even and odd numbers are statistically much more probable.

It’s important to note that this analysis assumes a random selection of integers. In real-world scenarios, number sets are often not formed randomly. For instance, the set of days in a week with an odd number of letters ({Monday, Wednesday, Friday}) is an all-“odd-length” set, but it wasn’t formed by random number selection. Similarly, the set of even page numbers in a book is intentionally all-even.

However, if we consider a scenario where we are simply picking a collection of numbers without any specific criteria related to parity, the mathematics clearly shows that all-even or all-odd sets become increasingly uncommon as the size of the set increases.

In conclusion, while all-even and all-odd number sets certainly exist, they are statistically less frequent than sets containing a mix of even and odd numbers. The probability of encountering such a set decreases exponentially with the number of elements in the set, highlighting the relative rarity of these seemingly simple numerical patterns in the broader landscape of possible number combinations.