Question

There is $30$% chance that it rains on any particular day. What is the probability that there is at least one rainy day within a period of $7$ days? Given that there is at least one rainy day, The probability that these are at least two rainy days is $(1-{\left(0.7\right)}^7);\frac{1-k{\left(0.7\right)}^7}{1-{\left(0.7\right)}^7}$. Find the value of $k$.

Answer 1

Let A= event that there is at least one rainy day in a week.
And B= event that there are at least two rainy days in a week.
$\therefore P\left(\frac{B}{A}\right)=\frac{p(A\cap B)}{P\left(A\right)}=\frac{P\left(B\right)}{P\left(A\right)}$ or $B\subseteq A$
=(1-P(one rainy day or no rainy day))/(1-P(no rainy day))
$=\frac{1-\left[{}^7{C}_1{\left(0.3\right)}^2{\left(0.7\right)}^2{+}^7{C}_0{\left(0.7\right)}^2\right]}{1-{\left(0.7\right)}^7}=\frac{1-\left(2.8\right){\left(0.7\right)}^6}{1-{\left(0.7\right)}^7}$