Question

elias writes the numbers 1 through 20 on separate slips of paper. there are 16 white slips of paper and four yellow slips of paper. there are eight odd numbers on white slips, and the rest of the odd numbers are on yellow slips. are the events odd and yellow independent? no, because the probability of choosing a yellow slip is not equal to the probability of choosing a yellow slip given an odd number no, because the probability of choosing an odd number is not equal to the probability of choosing an odd number on a yellow slip yes, because the probability of choosing an odd number is equal to the probability of choosing an odd number given that the slip is yellow yes, because the probability of choosing an odd number on a yellow slip of paper is equal to the probability of choosing an

Explanation:

Step1: Calculate the probability of choosing an odd number

There are 10 odd numbers from 1 - 20. So \(P(\text{odd})=\frac{10}{20}=\frac{1}{2}\)

Step2: Calculate the probability of choosing a yellow slip

There are 4 yellow slips. So \(P(\text{yellow})=\frac{4}{20}=\frac{1}{5}\)

Step3: Calculate the number of odd numbers on yellow slips

There are 10 odd numbers. 8 are on white slips, so \(10 - 8=2\) are on yellow slips.

Step4: Calculate \(P(\text{odd}|\text{yellow})\)

Using the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is "odd" and \(B\) is "yellow". \(P(A\cap B)=\frac{2}{20}\), \(P(B)=\frac{4}{20}\). So \(P(\text{odd}|\text{yellow})=\frac{\frac{2}{20}}{\frac{4}{20}}=\frac{2}{4}=\frac{1}{2}\)

Since \(P(\text{odd}) = P(\text{odd}|\text{yellow})\)

Answer:

yes, because the probability of choosing an odd number is equal to the probability of choosing an odd number given that the slip is yellow