Question

Let \( U = \{1, 2, 3, 4, 5\} \). A subset \( S \) is chosen uniformly at random from the non-empty subsets of \( U \). What is the probability that \( S \) does NOT have two consecutive elements?

• A. 9/31
• B. 10/31
• C. 11/31
• D. 12/31

Hint:

To find the probability of selecting subsets with a condition, count the number of favorable outcomes and divide by the total number of possible outcomes.

Answer 1

The Correct Option is D

Step 1: Understanding the problem.
We are given a set \( U = \{1, 2, 3, 4, 5\} \), and we need to find the probability that a subset \( S \) chosen from its non-empty subsets does not contain two consecutive elements.
Step 2: Count the valid subsets.
The valid subsets are those where no two consecutive elements are chosen from \( U \). The number of such subsets can be found by considering the binary representation of selecting elements. The valid subsets are:
\( \{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{1, 3\}, \{1, 4\}, \{1, 5\}, \{2, 4\}, \{2, 5\}, \{1, 3, 5\} \). Thus, there are 10 valid subsets.
Step 3: Total number of non-empty subsets.
The total number of non-empty subsets of \( U \) is \( 2^5 - 1 = 31 \), as each element can either be included or not, except the empty set.
Step 4: Conclusion.
The probability that \( S \) does not have two consecutive elements is \( \frac{10}{31} \), so the correct answer is (B) 10/31.