Question

If the probability of rain on any given day in Pune city is $50\%$. What is the probability that it rains on exactly 3 days in a 5 day period?

Answer 1

We have given the probability of rain on any given day in Pune city is $50\%$. We calculate the total number of possible outcomes. Then, we calculate the number of ways to choose 3 days in a 5 day period. Then, we calculate the probability by using the general formula of probability, which is given by
$P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$
Where, A is an event,
$n\left( E \right)=$ Number of favorable outcomes and $n\left( S \right)=$ number of total possible outcomes

** Complete step-by-step answer :**
We have given that the probability of rain on any given day in Pune city is $50\%$.
We have to find the probability that it rains on exactly 3 days in a 5 day period.
Now, as given in the question probability of rain on any given day in Pune city is $50\%$. It means the probability of no rain is also $50\%$.
We can write it as Probability of rain on a day is $\dfrac{1}{2}$.
Also, probability of no rain is $\dfrac{1}{2}$.
Now, the number of ways to choose exactly 3 days in a 5 day period will be
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
Where, $n=$ number of items/objects
And $r=$ number of items/objects being chosen at a time
So, number of ways to choose exactly 3 days in a 5 day period will be

& {}^{5}{{C}_{3}}=\dfrac{5!}{3!\left( 5-3 \right)!} \\\ & {}^{5}{{C}_{3}}=\dfrac{5\times 4\times 3!}{3!\left( 2 \right)!} \\\ & {}^{5}{{C}_{3}}=\dfrac{5\times 4}{2\times 1} \\\ & {}^{5}{{C}_{3}}=\dfrac{20}{2} \\\ & {}^{5}{{C}_{3}}=10 \\\ \end{aligned}$$ Now, the probability that it rains on exactly 3 days in a 5 day period will be $$\begin{aligned} & \Rightarrow {}^{5}{{C}_{3}}\times {{\left( \dfrac{1}{2} \right)}^{3}}\times {{\left( \dfrac{1}{2} \right)}^{2}} \\\ & \Rightarrow 10\times \dfrac{1}{8}\times \dfrac{1}{4} \\\ & \Rightarrow \dfrac{10}{32} \\\ & =\dfrac{5}{16} \\\ \end{aligned}$$ So, the probability that it rains on exactly 3 days in a 5 day period is $\dfrac{5}{16}$. **Note** : Alternatively we can solve this question by following way- Total number of days we have is $5$. So, the total number of possible outcomes will be ${{2}^{5}}=32$ Number of ways to choose 3 days in a period of 5 day will be $=10$ (As calculated above) Now, the probability that it rains on exactly 3 days in a 5 day period will be $P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$ $\begin{aligned} & P\left( A \right)=\dfrac{10}{32} \\\ & P\left( A \right)=\dfrac{5}{16} \\\ \end{aligned}$