Answer
a) Probability of a flood two years in a row is $0.04$.
b) Probability of a flood for three consecutive years is $0.008$.
c) The probability of no flooding for four consecutive years is $0.4096$.
Work Step by Step
(a)
We know that the probability of a flood in a given year in a region is,
${{\text{P}}_{\text{One year}}}\text{ = 0}\text{.2}$
And the probability of no flooding in a given year in a region is,
$\begin{align}
& {{\text{P}}_{\text{No flooding}}}\text{ = 1}-{{\text{P}}_{\text{One year}}}\text{ } \\
& \text{= 1}-\text{0}\text{.2} \\
& \text{= 0}\text{.8}
\end{align}$
Then, the probability of a flood two years in a row,
$\begin{align}
& {{\text{P}}_{\text{Two years }}}={{\text{P}}_{\text{One year}}}\times \text{ }{{\text{P}}_{\text{One year}}} \\
& \text{= 0}\text{.2 }\times \text{ 0}\text{.2} \\
& \text{= 0}\text{.04}
\end{align}$
(b)
We know that the probability of a flood for three consecutive years is
$\begin{align}
& {{\text{P}}_{\text{Three years }}}={{\text{P}}_{\text{One year}}}\times \text{ }{{\text{P}}_{\text{One year}}}\times \text{ }{{\text{P}}_{\text{One year}}} \\
& \text{= 0}\text{.2 }\times \text{ 0}\text{.2 }\times \text{ 0}\text{.2} \\
& \text{= 0}\text{.008}
\end{align}$
(c)
We know that the probability of no flooding for four consecutive years is
$\begin{align}
& {{\text{P}}_{\text{Four years no flooding}}}\text{ = }{{\text{P}}_{\text{No flooding}}}\times {{\text{P}}_{\text{No flooding}}}\times {{\text{P}}_{\text{No flooding}}}\times {{\text{P}}_{\text{No flooding}}} \\
& =0.8\times 0.8\times 0.8\times 0.8 \\
& =0.4096
\end{align}$
