Question

amdm
binomial theorem hw

1. on any day, the probability that it rains on a given day is 10%. over the next 3 days, what is the probability that it rains 2 days or more?

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \), where \( n \) is the number of trials, \( k \) is the number of successes, \( p \) is the probability of success on a single trial, and \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \). Here, \( n = 3 \) (days), \( p = 0.1 \) (probability of rain), and we need \( P(X\geq2)=P(X = 2)+P(X = 3) \).

Step2: Calculate \( P(X = 2) \)

First, calculate \( \binom{3}{2}=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!×1}=3 \). Then, \( p^k=(0.1)^2 = 0.01 \), \( (1 - p)^{n - k}=(0.9)^{1}=0.9 \). So \( P(X = 2)=3\times0.01\times0.9 = 0.027 \).

Step3: Calculate \( P(X = 3) \)

Calculate \( \binom{3}{3}=\frac{3!}{3!(3 - 3)!}=1 \). Then, \( p^k=(0.1)^3 = 0.001 \), \( (1 - p)^{n - k}=(0.9)^{0}=1 \). So \( P(X = 3)=1\times0.001\times1 = 0.001 \).

Step4: Sum the probabilities

\( P(X\geq2)=P(X = 2)+P(X = 3)=0.027 + 0.001=0.028 \).

Answer:

\( 0.028 \) (or 2.8%)