Question

A number is picked randomly from a given set of numbers. What is the probability that it is divisible by 3?
Statement 1: The set contains 10 consecutive integers
Statement 2: The first number of the set is divisible by 3

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We are asked to find the probability that a randomly picked number from a given set of numbers is divisible by 3. The two statements provided are:
- Statement 1: The set contains 10 consecutive integers.
- Statement 2: The first number of the set is divisible by 3.

Step 2: Analyze Statement 1.
Statement 1 tells us that the set contains 10 consecutive integers. The probability of selecting a number divisible by 3 depends on how many numbers divisible by 3 are in the set of 10 consecutive numbers.

In any set of 3 consecutive integers, exactly one of them is divisible by 3. Therefore, in a set of 10 consecutive integers, there will be 3 numbers divisible by 3 (since 10 ÷ 3 gives 3 full sets of 3 numbers, with 1 extra number from the remaining 1 number). Hence, the probability of selecting a number divisible by 3 is:
\[ \frac{3}{10} \] Therefore, Statement 1 alone is sufficient to answer the question.

Step 3: Analyze Statement 2.
Statement 2 tells us that the first number of the set is divisible by 3. This is useful, but it does not change the fact that we have 10 consecutive integers. Even if the first number is divisible by 3, the remaining numbers will still follow the same pattern, and there will still be 3 numbers divisible by 3 in the set. Hence, Statement 2 alone does not provide any additional information beyond Statement 1.

Step 4: Conclusion.
Statement 1 alone is sufficient to answer the question. The probability that a randomly picked number from the set is divisible by 3 is \( \frac{3}{10} \).

Final Answer:
The correct option is (A): statement (1) alone is sufficient to answer the question.