A hydraulic structure has four gates which operate independently. The probability of failure of each gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is

The correct answer is $\boxed{\text{C) 0.040}}$.

The probability of two events happening independently is the product of their probabilities. In this case, the probability of gate 2 failing is 0.2, and the probability of gate 3 failing is also 0.2. So, the probability of both gates 2 and 3 failing is $0.2 \times 0.2 = 0.04$.

Option A is incorrect because it is the probability of all four gates failing. The probability of all four gates failing is $0.2 \times 0.2 \times 0.2 \times 0.2 = 0.0016$.

Option B is incorrect because it is the probability of gate 2 failing given that gate 1 has failed. The probability of gate 2 failing given that gate 1 has failed is not the same as the probability of both gates 2 and 3 failing.

Option D is incorrect because it is the probability of gate 3 failing given that gate 1 has failed. The probability of gate 3 failing given that gate 1 has failed is not the same as the probability of both gates 2 and 3 failing.