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A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If \(\mathrm{n}\) of the bulbs in box are defective, what is the value of \(\mathrm{n}\)?
We need to find the exact value of n, where n is the number of defective bulbs in a box of 10 bulbs.
For sufficiency, we need information that narrows down n to exactly one value from {0, 1, 2, 3, 4}.
Since we're dealing with just 5 possible values, we can test each one directly against any probability constraint. This approach is often faster than setting up and solving equations.
Statement 1 tells us: The probability that both bulbs drawn will be defective is \(\frac{1}{15}\).
Let's test which value of n gives us this probability. When drawing 2 bulbs from 10, there are \(\mathrm{C(10,2) = 45}\) total ways.
Only n = 3 produces the probability of \(\frac{1}{15}\). [STOP - Sufficient!]
Statement 1 is sufficient.
This eliminates choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The probability that one bulb will be defective and the other non-defective is \(\frac{7}{15}\).
Again, with 45 total ways to draw 2 bulbs, let's test our possible values.
For each n, the number of ways to draw 1 defective and 1 non-defective = \(\mathrm{n × (10-n)}\)
Only n = 3 produces the probability of \(\frac{7}{15}\). [STOP - Sufficient!]
Statement 2 is sufficient.
Since each statement alone is sufficient to determine that n = 3, the answer is D.
Answer Choice D: "Each statement alone is sufficient."