Question

Let M be a randomly chosen non-empty subset of \{1,2,3,..,2026\. Which of the following is a probability that product all the elements of M is even.}

• A. $\frac{2^{1013} (2^{1013}-1)}{(2^{2026}-1)}$
• B. $\frac{2^{1013} (2^{1013}-1)}{2^{2026}}$
• C. $\frac{1}{(2^{2026}-1)}$
• D. $\frac{2^{1013}}{2^{2026}}$

Hint:

For probability questions involving "at least one" or properties like "even product", using the complement is usually the simplest approach. An even product requires at least one even number. The complement is that there are *zero* even numbers, meaning all numbers are odd.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are asked to find the probability that the product of elements in a randomly chosen non-empty subset of S = \{1, 2, ..., 2026\} is even. It's often easier to calculate the probability of the complement event and subtract it from 1.
Step 2: Key Formula or Approach:
P(Product is Even) = 1 - P(Product is Odd).
The product of a set of integers is odd if and only if all the integers in the set are odd. So, we need to count the number of non-empty subsets that contain only odd numbers.
Step 3: Detailed Explanation:
First, let's analyze the set S = \{1, 2, ..., 2026\}.
- Total number of elements in S = 2026.
- The set of odd numbers in S is \{1, 3, 5, ..., 2025\}. Number of odd numbers = 1013.
- The set of even numbers in S is \{2, 4, 6, ..., 2026\}. Number of even numbers = 1013.
Now, let's find the total number of possible outcomes.
- The total number of subsets of S is $2^{2026}$.
- The question specifies a "non-empty subset", so the total number of outcomes is $2^{2026} - 1$.
Next, let's find the number of favorable outcomes for the complement event (product is odd).
- A subset's product is odd only if it contains exclusively odd numbers. - The number of subsets that can be formed using only the 1013 odd numbers is $2^{1013}$. - Since the subset must be non-empty, we exclude the empty set. So, the number of non-empty subsets with an odd product is $2^{1013} - 1$.
Now, calculate the probability of the product being odd: \[ P(\text{Product is Odd}) = \frac{\text{Number of subsets with odd product}}{\text{Total number of non-empty subsets}} = \frac{2^{1013} - 1}{2^{2026} - 1} \] Finally, calculate the probability of the product being even: \[ P(\text{Product is Even}) = 1 - P(\text{Product is Odd}) \] \[ P(\text{Product is Even}) = 1 - \frac{2^{1013} - 1}{2^{2026} - 1} \] \[ P(\text{Product is Even}) = \frac{(2^{2026} - 1) - (2^{1013} - 1)}{2^{2026} - 1} \] \[ P(\text{Product is Even}) = \frac{2^{2026} - 1 - 2^{1013} + 1}{2^{2026} - 1} \] \[ P(\text{Product is Even}) = \frac{2^{2026} - 2^{1013}}{2^{2026} - 1} \] Factor out $2^{1013}$ from the numerator: \[ P(\text{Product is Even}) = \frac{2^{1013}(2^{1013} - 1)}{2^{2026} - 1} \] Step 4: Final Answer:
The calculated probability matches option (A).