Question

determining if events are independent
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 odd number

Explanation:

Step1: Calculate probability of odd number

Total numbers \(n = 20\). Odd numbers from \(1\) - \(20\): \(1,3,\cdots,19\), so \(n(\text{odd})=10\). Probability \(P(\text{odd})=\frac{10}{20}=\frac{1}{2}\).

Step2: Calculate probability of yellow slip

Total slips \(n = 20\), yellow slips \(n(\text{yellow}) = 4\). Probability \(P(\text{yellow})=\frac{4}{20}=\frac{1}{5}\).

Step3: Calculate probability of odd - numbered yellow slips

Odd - numbered yellow slips: \(10 - 8=2\). Probability \(P(\text{odd and yellow})=\frac{2}{20}=\frac{1}{10}\).

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

By formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), so \(P(\text{odd}|\text{yellow})=\frac{\frac{1}{10}}{\frac{1}{5}}=\frac{1}{2}\).

Since \(P(\text{odd}) = P(\text{odd}|\text{yellow})=\frac{1}{2}\), events “odd” and “yellow” are independent.

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.