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A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that x∈A is
- a)64/127
- b)63/128
- c)1/2
- d)31/128
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A set S contains 7 elements. A non-empty subset A of S and an element ...
Total non empty subsects = 27 –1 = 127 Let x ∈ S also present in A
So no. of A's containg x = 26
So no. of A's containg x = 26
Most Upvoted Answer
A set S contains 7 elements. A non-empty subset A of S and an element ...
Understanding the Problem
To find the probability that a randomly chosen element x from set S is included in a randomly chosen non-empty subset A, we need to analyze the combinations possible.
Elements in Set S
- Let set S contain 7 elements.
- The total number of subsets of S is 2^7, which equals 128.
- This includes the empty subset, so the number of non-empty subsets is 128 - 1 = 127.
Choosing Element x
- We can choose any of the 7 elements from set S as x.
- Each element has a chance of being in any non-empty subset A.
Non-empty Subset Inclusion
- For each non-empty subset A, consider how many of the 127 subsets contain a specific element x.
- If x is included in A, the remaining elements can either be included or excluded from A.
Counting Subsets with x
- For the 6 other elements (excluding x), each can either be included or not, giving us 2^6 = 64 ways.
- Therefore, the number of non-empty subsets A that include x is 64, as we must also consider that A cannot be empty.
Calculating the Probability
- The probability that x is in A can be calculated as follows:
Probability = (Number of non-empty subsets containing x) / (Total number of non-empty subsets)
= 64 / 127.
Conclusion
Thus, the probability that a randomly chosen element x from set S is included in a randomly chosen non-empty subset A is:
Correct Answer: 64/127 (Option 'A')
To find the probability that a randomly chosen element x from set S is included in a randomly chosen non-empty subset A, we need to analyze the combinations possible.
Elements in Set S
- Let set S contain 7 elements.
- The total number of subsets of S is 2^7, which equals 128.
- This includes the empty subset, so the number of non-empty subsets is 128 - 1 = 127.
Choosing Element x
- We can choose any of the 7 elements from set S as x.
- Each element has a chance of being in any non-empty subset A.
Non-empty Subset Inclusion
- For each non-empty subset A, consider how many of the 127 subsets contain a specific element x.
- If x is included in A, the remaining elements can either be included or excluded from A.
Counting Subsets with x
- For the 6 other elements (excluding x), each can either be included or not, giving us 2^6 = 64 ways.
- Therefore, the number of non-empty subsets A that include x is 64, as we must also consider that A cannot be empty.
Calculating the Probability
- The probability that x is in A can be calculated as follows:
Probability = (Number of non-empty subsets containing x) / (Total number of non-empty subsets)
= 64 / 127.
Conclusion
Thus, the probability that a randomly chosen element x from set S is included in a randomly chosen non-empty subset A is:
Correct Answer: 64/127 (Option 'A')
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A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that x∈A isa)64/127b)63/128c)1/2d)31/128Correct answer is option 'A'. Can you explain this answer?
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