Probability That a Random Subset of {1, 2, 3, 4, 5} Has No Two Consecutive Elements

Understanding the Random Subset Probability Problem

This question combines the concepts of subsets, combinatorics, and probability. We are given the universal set:

U = {1, 2, 3, 4, 5}

A non-empty subset S is selected uniformly at random. The phrase “uniformly at random” means that every non-empty subset has the same probability of being selected. Therefore, the required probability can be calculated using the standard formula:

Probability = Number of favorable non-empty subsets / Total number of non-empty subsets

The favorable subsets are those that do not contain any pair of consecutive elements. For example, {1, 3} is allowed because 1 and 3 are not consecutive, while {1, 2} is not allowed because 1 and 2 are consecutive.

Similarly, {1, 3, 5} is allowed because none of its elements are consecutive. However, a subset such as {1, 3, 4} is not allowed because it contains the consecutive pair 3 and 4.

What Does “No Two Consecutive Elements” Mean?

Two integers are consecutive when their difference is 1. In the set U = {1, 2, 3, 4, 5}, the consecutive pairs are:

(1, 2), (2, 3), (3, 4), and (4, 5)

Therefore, a favorable subset cannot contain both elements from any of these pairs.

For instance, the subset {2, 4} is valid because the difference between 2 and 4 is 2. The subset {1, 4, 5}, however, is invalid because 4 and 5 are consecutive, even though 1 is not consecutive with either of them.

The condition must be satisfied by every pair of elements present in the subset.

Step-by-Step Solution

Step 1: Calculate the Total Number of Non-Empty Subsets

A set containing n elements has a total of:

2ⁿ subsets

The given set U contains five elements. Therefore, the total number of subsets is:

2⁵ = 32

These 32 subsets include the empty set ∅. However, the question specifically states that S is selected from the non-empty subsets of U. Therefore, the empty set must be excluded.

Hence:

Total number of non-empty subsets = 32 − 1

Therefore:

Total number of possible subsets = 31

Step 2: Count the Valid One-Element Subsets

Every one-element subset automatically satisfies the condition because a subset containing only one element cannot contain a pair of consecutive elements.

The one-element subsets are:

{1}, {2}, {3}, {4}, {5}

Therefore:

Number of valid one-element subsets = 5

Step 3: Count the Valid Two-Element Subsets

Now we need to count the two-element subsets in which the selected integers are not consecutive.

The valid two-element subsets are:

{1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 5}

Each of these subsets contains two elements whose difference is greater than 1. Therefore, none of them contains consecutive integers.

Hence:

Number of valid two-element subsets = 6

Step 4: Count the Valid Three-Element Subsets

We now consider subsets containing three elements. To avoid consecutive elements, the selected numbers must be separated from one another.

For the set {1, 2, 3, 4, 5}, the only possible three-element subset with no two consecutive elements is:

{1, 3, 5}

The elements 1 and 3 are not consecutive, 3 and 5 are not consecutive, and 1 and 5 are also not consecutive.

Therefore:

Number of valid three-element subsets = 1

Step 5: Check Subsets with Four or Five Elements

It is impossible to select four elements from {1, 2, 3, 4, 5} without selecting at least one consecutive pair. Similarly, the complete five-element set certainly contains several consecutive pairs.

Therefore:

Number of valid four-element subsets = 0

and:

Number of valid five-element subsets = 0

Step 6: Calculate the Total Number of Favorable Subsets

The valid non-empty subsets consist of:

5 one-element subsets

6 two-element subsets

1 three-element subset

Therefore, the total number of favorable subsets is:

5 + 6 + 1 = 12

Hence:

Number of favorable subsets = 12

Step 7: Calculate the Required Probability

Using the probability formula:

Probability = Number of favorable outcomes / Total number of possible outcomes

we obtain:

Probability = 12/31

Therefore:

P(S has no two consecutive elements) = 12/31

Complete List of All Favorable Non-Empty Subsets

To verify the counting directly, all favorable subsets can be listed according to their sizes.

The five valid one-element subsets are:

{1}, {2}, {3}, {4}, {5}

The six valid two-element subsets are:

{1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 5}

The only valid three-element subset is:

{1, 3, 5}

Thus, the total number of favorable non-empty subsets is:

5 + 6 + 1 = 12

Since there are 31 non-empty subsets in total, the required probability is 12/31.

Alternative Solution Using a Combinatorial Formula

There is also a general formula for selecting k elements from {1, 2, …, n} such that no two selected elements are consecutive.

The number of ways is:

C(n − k + 1, k)

For the present problem, n = 5.

For one-element subsets:

C(5 − 1 + 1, 1) = C(5, 1) = 5

For two-element subsets:

C(5 − 2 + 1, 2) = C(4, 2) = 6

For three-element subsets:

C(5 − 3 + 1, 3) = C(3, 3) = 1

Therefore:

Total favorable subsets = 5 + 6 + 1

= 12

Hence:

Probability = 12/31

Why the Empty Set Is Not Counted

The empty set contains no consecutive elements, so mathematically it also satisfies the condition of having no two consecutive elements. However, the question clearly states that S is chosen from the non-empty subsets of U.

Therefore, the empty set is excluded from both the total number of possible subsets and the number of favorable subsets.

If the empty set had been allowed, there would be 13 favorable subsets out of 32 total subsets. But under the actual condition of the question, the correct values are 12 favorable subsets out of 31 possible non-empty subsets.

Detailed Analysis of Each Option

Option (A): 9/31

This option is incorrect because there are more than nine favorable subsets. The five singleton subsets and six valid two-element subsets alone already give 11 favorable subsets. Including the valid three-element subset {1, 3, 5} increases the total to 12.

Option (B): 10/31

This option is incorrect. A direct enumeration of all valid non-empty subsets gives 5 one-element subsets, 6 two-element subsets, and 1 three-element subset. Their total is 12, not 10.

Option (C): 11/31

This option is incorrect. The value 11 can arise if the valid three-element subset {1, 3, 5} is overlooked. The five singletons and six valid pairs give 11, but {1, 3, 5} also contains no consecutive elements and must be included.

Option (D): 12/31

This option is correct. There are exactly 12 favorable non-empty subsets and 31 non-empty subsets in total. Therefore, the required probability is:

12/31

Verification of the Final Result

The total number of non-empty subsets is:

2⁵ − 1 = 31

The number of favorable subsets is:

5 + 6 + 1 = 12

Therefore:

P(no two consecutive elements) = 12/31

This confirms that the correct answer is Option (D).

Final Answer

The probability that a uniformly chosen non-empty subset of U = {1, 2, 3, 4, 5} does not contain two consecutive elements is 12/31.

Correct Option: (D) 12/31