Question

On an average if it rains on $10$ days in every $30$, the probability that there will be rain on at least three days of a given week is

• A. ${}_3^{7}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^3$
• B. $1{-}_0^{7}C{\left(\frac{2}{3}\right)}^7{-}_1^{7}C{\left(\frac{2}{3}\right)}^6{\left(\frac{1}{3}\right)}^1{-}_2^{7}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^2$
• C. ${}_3^{7}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^3{+}_4^{7}C{\left(\frac{2}{3}\right)}^3{\left(\frac{1}{3}\right)}^4{+}_5^{7}C{\left(\frac{2}{3}\right)}^2{\left(\frac{1}{3}\right)}^5$
• D. ${}_0^{7}C{\left(\frac{2}{3}\right)}^7{+}_1^{7}C{\left(\frac{2}{3}\right)}^6{\left(\frac{1}{3}\right)}^1{+}_2^{7}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^2$

Answer 1

The correct option is B $1{-}_0^{7}C{\left(\frac{2}{3}\right)}^7{-}_1^{7}C{\left(\frac{2}{3}\right)}^6{\left(\frac{1}{3}\right)}^1{-}_2^{7}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^2$

Probability of raining is $\frac{10}{30}$
$=\frac{1}{3}$
Probability of not raining is $\frac{2}{3}$
Now the probability of rainfall on 3 or more days out of 7 days is

${\sum }_{k=3}^7{}^7{C}_k\left(\frac{1}{3}{)}^k\right(\frac{2}{3}{)}^{7-k}$