If the risk of a flood occurring in the next 10 years is accepted to 10%, then the return period for design should be A. 1 + 0.9°10 B. 1 – 0.9°10 C. $$\frac{1}{{1 – {{0.9}^ \circ }10}}$$ D. $$\frac{1}{{1 + {{0.9}^ \circ }10}}$$

The correct answer is $\boxed{\frac{1}{{1 – {{0.9}^ \circ }10}}}$.

The return period of a flood is the average number of years between floods of a given magnitude or greater. A 10% risk of a flood occurring in the next 10 years means that there is a 10% chance of a flood occurring in any given year. In other words, the flood has a 10% chance of occurring once every 10 years, on average.

The return period can be calculated using the following formula:

$T = \frac{1}{{1 – {{p}^ \circ }r}}$

where:

• $T$ is the return period
• $p$ is the probability of the event occurring in any given year
• $r$ is the number of years

In this case, $p = 0.1$ and $r = 10$. Substituting these values into the formula, we get:

$T = \frac{1}{{1 – {{0.1}^ \circ }10}} = \frac{1}{{1 – 0.09090909090909091}} = 10.909090909090909$

Therefore, the return period for design is 10.9 years. This means that there is a 10% chance of a flood occurring once every 10.9 years, on average.

Option A is incorrect because it is the probability of the event occurring in any given year, not the return period.

Option B is incorrect because it is the probability of the event not occurring in any given year.

Option C is incorrect because it is the reciprocal of the probability of the event not occurring in any given year.

Option D is incorrect because it is the reciprocal of the probability of the event occurring in any given year.