Question

The probability of rain on any given day is 2/3 . What is the probability of at most one day of rain during the next three days?

Answer 1

Answer:

$\frac{7}{27}$

Step-by-step explanation:

Let the event of raining on a day be success.

Then probability of success, $p=\frac{2}{3}$

So, probability of failure is not raining on a day and is given as,

$q=1-p=1-\frac{2}{3}=\frac{3-2}{3}=\frac{1}{3}$

Now, number of days to check the probability of rain, $n=3$

Now, probability of raining at most one day means number of success is either 0 or 1.

Using Bernoulli's distribution for $X=0,1$

$P(X=0\textrm{ or }X=1)=_{0}^{3}\textrm{C}p^{0}q^{3-0}+_{1}^{3}\textrm{C}p^{1}q^{3-1}\\\\P(X=0\textrm{ or }X=1)=(\frac{1}{3})^{3}+3(\frac{2}{3})^{1}(\frac{1}{3})^{2}\\\\P(X=0\textrm{ or }X=1)=\frac{1}{27}+\frac{2}{9}\\\\P(X=0\textrm{ or }X=1)=\frac{1}{27}+\frac{6}{27}\\\\P(X=0\textrm{ or }X=1)=\frac{1+6}{27}=\frac{7}{27}$

Therefore, the probability of at most one day of rain during the next three days is $\frac{7}{27}$