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 odd number

Explanation:

Step1: Calculate total number of odd - numbers

Numbers from 1 to 20, odd - numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. So there are 10 odd - numbers.

Step2: Calculate probability of choosing an odd number

The total number of slips is 20. The probability of choosing an odd number, $P(O)=\frac{10}{20}=\frac{1}{2}$.

Step3: Calculate number of odd numbers on yellow slips

There are 8 odd numbers on white slips, so the number of odd numbers on yellow slips is $10 - 8=2$.

Step4: Calculate probability of choosing an odd number given the slip is yellow

There are 4 yellow slips. The probability of choosing an odd number given the slip is yellow, $P(O|Y)=\frac{2}{4}=\frac{1}{2}$.

Step5: Check for independence

Two events A and B are independent if $P(A)=P(A|B)$. Here, event A is choosing an odd number and event B is choosing a yellow slip. Since $P(O)=\frac{1}{2}$ and $P(O|Y)=\frac{1}{2}$, the 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