Question

There is $30$ chance that it rains on any particular day. Given that there is at least one rainy day in a week, what is the probability that there are at least two rainy days in the week?

• A. $\frac{1-4(0.7{)}^7}{1-(0.7{)}^7}$
• B. $\frac{4(0.7{)}^7}{1-(0.7{)}^7}$
• C. $\frac{1-(0.7{)}^7}{1-4.(0.7{)}^7}$
• D. $\frac{(0.7{)}^7}{1-4.(0.7{)}^7}$

Answer 1

The correct option is A $\frac{1-4(0.7{)}^7}{1-(0.7{)}^7}$

Let $A$ be the event that there is at least one rainy day in a week.
Let $B$ be the event that there are at least two rainy days in a week

$\therefore P\left(\frac{B}{A}\right)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P\left(B\right)}{P\left(A\right)}$ as $B\subseteq A$

$=\frac{(1-P(\text{one rainy day or no rainy day}\left)\right)}{(1-P(\text{no rainiy day}\left)\right)}$

$=\frac{1-\left[{}^7{C}_1{\left(0.3\right)}^1{\left(0.7\right)}^6{+}^7{C}_0{\left(0.7\right)}^7\right]}{1-{\left(0.7\right)}^7}=\frac{1-\left(4\right){\left(0.7\right)}^7}{1-{\left(0.7\right)}^7}$