What we see: a Data Sufficiency probability question. Joe draws a tile \(n\), builds Sequence \(Q = n, n+1, \dots, n+9\), then picks one number from \(Q\). We need the share of the 10 numbers in \(Q\) that are multiples of 3.
Step 1. Count the multiples of 3 in any 10 numbers in a row.
What we see: \(Q\) is always 10 consecutive integers.
What it means: 10 numbers in a row hold either 3 or 4 multiples of 3. They hold 4 only when the first number is itself a multiple of 3, because then \(n, n+3, n+6, n+9\) all are. Example: 3 to 12 holds 3, 6, 9, 12. But 4 to 13 holds only 6, 9, 12.
What we do and why: the probability is \(\tfrac{4}{10}\) when \(n\) is a multiple of 3 and \(\tfrac{3}{10}\) when it is not. So the real question is simply: is the drawn number a multiple of 3? We never need \(n\) itself.
Step 2. Statement 1.
What we see: the last number, \(n+9\), is a prime less than 20.
What it means: \(n\) is at least 1, so \(n+9\) is at least 10. The primes from 10 to 19 are 11, 13, 17 and 19, so \(n\) is 2, 4, 8 or 10. None of those is a multiple of 3.
What we do and why: every case gives \(\tfrac{3}{10}\). One answer, so statement 1 is sufficient.
Step 3. Statement 2.
What we see: \(x \le 10\), so the tiles go up to 10 at most.
What it means: Joe could still draw 3 (probability \(\tfrac{4}{10}\)) or 1 (probability \(\tfrac{3}{10}\)). The statement limits the tiles, not which one was drawn.
What we do and why: two different answers, so statement 2 is not sufficient.
The trap: C or E, from thinking you must know \(x\), the number of tiles, to find any probability here. The tiles only limit which \(n\) is possible. The probability asked about is the pick from \(Q\), and statement 1 settles that by itself.
Faster way: once you see that only "is \(n\) a multiple of 3?" matters, list the four possible \(n\) values from statement 1 and check them. That takes under a minute.
Takeaway: When a probability depends on remainders, shrink the question to "which remainder?" first. A statement then only has to fix the remainder, not the exact number.
Answer: A
