Question

use the provided information to answer each part of the question below. in the last 20 days it has rained 13 times. the odds that alexis, a kindergarten teacher, takes her umbrella when she leaves the house is always 1:3. a. state the odds against rain today. b. determine the probability that it will rain and alexis will be carrying her umbrella. c. explain whether the odds in the information box are independent or dependent.

Explanation:

Step1: Calculate odds against rain

The probability of rain in the last 20 days is $P(\text{rain})=\frac{13}{20}$. The probability of no - rain is $P(\text{no rain}) = 1-\frac{13}{20}=\frac{7}{20}$. The odds against rain is $\frac{P(\text{no rain})}{P(\text{rain})}=\frac{\frac{7}{20}}{\frac{13}{20}}=\frac{7}{13}$, or $7:13$.

Step2: Calculate probability of rain and umbrella

The probability that it rains is $P(\text{rain})=\frac{13}{20}$. The odds that Alexis takes her umbrella is $1:3$, so the probability that she takes her umbrella is $P(\text{umbrella})=\frac{1}{1 + 3}=\frac{1}{4}$. Since the events of rain and Alexis taking an umbrella are assumed to be independent (no information suggesting otherwise), the probability that it rains and she takes her umbrella is $P(\text{rain and umbrella})=P(\text{rain})\times P(\text{umbrella})=\frac{13}{20}\times\frac{1}{4}=\frac{13}{80}$.

Step3: Determine independence

The odds of rain in the last 20 days and the odds that Alexis takes her umbrella are independent. The number of times it rained in the last 20 days has no bearing on whether Alexis takes her umbrella. Her decision to take an umbrella is a fixed ratio ($1:3$) regardless of the past rain - data.

Answer:

a. $7:13$
b. $\frac{13}{80}$
c. The odds are independent. The number of times it rained in the last 20 days does not affect Alexis's decision to take her umbrella, which is always in the ratio of $1:3$.