Question at position 7 CAEBD

Question

Question at position 7 CAEBD

BlackTom AI · Accepted Answer

Here is the question restated and the options analyzed in detail.
Question: Which of the following is the probability that a randomly selected student does not send an average of more than 50 texts per day, given that they own a smartphone?
Answer options: A. 0.4  B. 0.333  C. 0.294  D. 0.2  E. 0.133

First, interpret the Venn diagram. The left circle S represents students who own a smartphone, the right circle T represents students who send an average of more than 50 texts per day. The regions have the following probabilities:
- S only (own smartphone, not more than 50 texts): 0.25
- T only (more than 50 texts, not necessarily on smartphone): 0.1
- Intersection (own smartphone and more than 50 texts): 0.5
- Outside both (neither condition): 0.15

We are asked for P(not T | S). This is the probability that a student does not send more than 50 texts, given that they own a smartphone. In probability notation, P(not T | S) = P(S ∩ not T) / P(S).

Now evaluate each component:
- P(S) is the probability a student owns a smartphone, regardless of texting. From the diagram, S includes S only and the intersection: 0.25 + 0.5 = 0.75.
- P(S ∩ not T) is the portion of S that does not belong to T. This is precisely the S only region, which equals 0.25.

Compute the conditional probability: P(not T | S) = 0.25 / 0.75 = 1/3 ≈ 0.333.

Option-by-option reasoning:
- Option A: 0.4. This would require P(S ∩ not T) to be 0.4 times P(S), i.e., 0.4 × 0.75 = 0.30 for P(S ∩ not T), which contradicts the diagram where P(S ∩ not T) is 0.25. Therefore, A is incorrect.
- Option B: 0.333. This matches the calculated ratio 0.25 / 0.75 = 1/3. It aligns with the S-only region (0.25) over the total S region (0.75). Therefore, B is correct.
- Option C: 0.294. If this were correct, P(S ∩ not T) would have to be 0.294 × 0.75 ≈ 0.2205, which is not present in the diagram. This inconsistency makes C invalid.
- Option D: 0.2. This would imply P(S ∩ not T) = 0.2 × 0.75 = 0.15, but 0.15 is the probability of being outside both groups, not S ∩ not T. So D is incorrect.
- Option E: 0.133. This would require P(S ∩ not T) = 0.133 × 0.75 ≈ 0.0998, which again does not match any region given in the diagram. Hence E is incorrect.

Summary observations: The only region that represents the event S ∩ not T is the S-only portion with probability 0.25, and the total S region is 0.25 + 0.5 = 0.75. The conditional probability hence is 0.25 / 0.75 = 0.333, which corresponds to option B.

Question at position 7 CAEBD

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