Question

new york city typically has rain on about 16% of days in October what is the probability that it will rain on exactly 5 days in October and 15 days?

Answer 1

The number of rainy days, $X$ can be represented by a binomial distribution with $n=31$ trials (the number of days in the month of October), success probability $p=0.16$ (representing a rainy day) and failure probability $q=1-p=0.84$ .

Therefore,

$$P(5)=_{n}C_{5}p^5q^{n-5}=_{31}C_{5}(0.16)^5(0.84)^{26}\approx0.191$$ $$P(15)=_{n}C_{15}p^{15}q^{n-15}=_{31}C_{15}(0.16)^{15}(0.84)^{16}\approx0.0000213$$