Question

the probability it will rain on a given day in july is 15% .what is the probability that within one week it will rain no more than 3 days

Answer 1

Let's denote $P(X=k)$ the probability that it will rain k days a week. Since raining on particular day of the week doesn't depend on other days (events are independent), we can find this probability as

$P(X=k) = С(7, k) \cdot 0.15^k \cdot 0.85^{7-k}$

because we have k days when it will rain (with probability 0.15) and 7-k days with no rain( with probability 0.85). Binomial coefficient is here because there can be permutations. For example, when we count on what 3 days exactly there will be rain, we can choose those days in C(7,3) ways. Then the condition "it will rain no more than 3 days" can be written as

$P(X \leq3) = P(X=0) +P(X=1) + P(X=2) + P(X=3)$

$P(X \leq3) = 0.85^7 + 7 \cdot 0.15 \cdot 0.85^6 + 21 \cdot 0.15^2 \cdot 0.85^5 +35 \cdot0.15^3 \cdot 0.85^4 = 0.3206 +0.396+0.210+0.062 = 0.9886$

Answer: 0.9886