Question

A fish tank contains a number of fish, including 5 Fantails. If two fish are selected from the tank at random, what is the probability that both will be Fantails?
(1) The probability that the first fish chosen will be a Fantail is \(\frac 12\) .
(2) The probability that the second fish chosen will be a Fantail is \(\frac 49\) .

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.
• D. EACH statement ALONE is sufficient.

Hint:

Sometimes either statement contains enough information to reconstruct the entire situation — always check each one individually.

Answer 1

The Correct Option is D

From (1): \(P(\text{first Fantail}) = \frac{5}{N} = \frac{1}{2} \Rightarrow N = 10\). With 5 Fantails in 10 fish: \[ P(\text{both Fantails}) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9} \] So statement (1) alone is sufficient. From (2): \(P(\text{second Fantail}) = \frac{4}{9}\) means that after one fish was removed, there were 4 Fantails among 9 fish. This implies initially there were 5 Fantails among 10 fish. Probability both Fantails again = \(\frac{5}{10} \times \frac{4}{9} = \frac{2}{9}\). Thus statement (2) alone is also sufficient. Therefore, \(\boxed{\text{D}}\) is correct.