Question

The probability of rain today is 60 percent, and the independent probability of rain tomorrow is 75 percent. Calculate the probability that it will rain neither today nor tomorrow.
(a). 10%
(b). 15%
(c). 45%
(d). 55%
(e). 65%

Answer 1

Hint: Here, we have to apply the property of probability that the total probability is 1, but here it is given in percentage so we can take 100% as 1.If two events A and B are independent then $P(A\cap B)=P(A).P(B)$

Complete step-by-step answer:
Here, let us consider two events A and B where:
A: It will rain today.
B: It will rain tomorrow.
We know that the total probability is 1. Here, the probability is given in percentage.
Therefore, we can take $100\%=1$
In the question, it is given that probability of rain today is 60% and probability of rain tomorrow is 75%. i.e.
$\begin{align}
  & P(A)=60\% \\
 & P(A)=\dfrac{60}{100} \\
\end{align}$
By cancellation we get,
$\begin{align}
  & P(A)=\dfrac{6}{10} \\
 & P(A)=0.6 \\
\end{align}$
Similarly, we have,
$\begin{align}
  & P(B)=75\% \\
 & P(B)=\dfrac{75}{100} \\
\end{align}$
By cancellation we obtain:
$P(B)=0.75$
Next, we have to calculate the probability that it will rain neither today nor tomorrow.
For that we have to consider another two events $\bar{A}$ and $\bar{B}$ where,
$\bar{A}$: It will not rain today
$\bar{B}$: It will not rain tomorrow.
$\bar{A}\cap \bar{B}$: It will rain neither today nor tomorrow.
Now, we have to find $P(\bar{A}\cap \bar{B})$.
Since, $\bar{A}$ and $\bar{B}$ are independent sets, we can say that:
$P(\bar{A}\cap \bar{B})=P(\bar{A}).P(\bar{B})$ ……. (1)
Hence, we can write:
$P(\bar{A})=P$(no rain today)
We know that the total probability is 1. Therefore, we can write:
$P(\bar{A})=1-P$(rain today)
$\begin{align}
  & P(\bar{A})=1-P(A) \\
 & P(\bar{A})=1-0.6 \\
 & P(\bar{A})=0.4 \\
\end{align}$Similarly, we can write:
$P(\bar{B})=P$(no rain tomorrow)
$P(\bar{B})=1-P$(rain tomorrow)
$\begin{align}
  & P(\bar{B})=1-P(B) \\
 & P(\bar{B})=1-0.75 \\
 & P(\bar{B})=0.25 \\
\end{align}$
Substituting the values of $P(\bar{A})$ and $P(\bar{B})$ in equation (1) we obtain:
$\begin{align}
  & P(\bar{A}\cap \bar{B})=0.4\times 0.25 \\
 & P(\bar{A}\cap \bar{B})=0.1 \\
 & P(\bar{A}\cap \bar{B})=\dfrac{10}{100} \\
 & P(\bar{A}\cap \bar{B})=10\% \\
\end{align}$
Hence, the probability that it will rain neither today nor tomorrow is 10%.
Therefore, the correct answer for this question is option (a).

Note:
Here, the event for neither rain today nor tomorrow is the complement of $A\cup B$, where, $A\cup B$ is the event that it will rain today or tomorrow and $\overline{A\cup B}=\overline{A}\cap \overline{B}$. Since, A and B are independent, so you can calculate $P(A\cap B)$ and then find the value of $P(A\cup B)$ and eventually you can find the value of $P\left( \overline{A}\cap \overline{B} \right)$.