elias writes the numbers 1 through 20 on separate slips of paper. there are 16 white slips of paper and four…

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?\nno, because the probability of choosing a yellow slip is not equal to the probability of choosing a yellow slip given an odd number\nno, because the probability of choosing an odd number is not equal to the probability of choosing an odd number on a yellow slip\nyes, because the probability of choosing an odd number is equal to the probability of choosing an odd number given that the slip is yellow\nyes, because the probability of choosing an odd number on a yellow slip of paper is equal to the probability of choosing an odd number

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?\nno, because the probability of choosing a yellow slip is not equal to the probability of choosing a yellow slip given an odd number\nno, because the probability of choosing an odd number is not equal to the probability of choosing an odd number on a yellow slip\nyes, because the probability of choosing an odd number is equal to the probability of choosing an odd number given that the slip is yellow\nyes, because the probability of choosing an odd number on a yellow slip of paper is equal to the probability of choosing an odd number

Answer

Explanation:

Step1: Calculate probability of yellow

There are 4 yellow slips out of 20 total slips. So $P(Y)=\frac{4}{20}=\frac{1}{5}$.

Step2: Calculate number of odd - yellow slips

There are 10 odd numbers from 1 - 20. 8 odd numbers are on white slips, so 2 odd numbers are on yellow slips.

Step3: Calculate probability of odd - yellow

$P(O\cap Y)=\frac{2}{20}=\frac{1}{10}$.

Step4: Calculate probability of odd

$P(O)=\frac{10}{20}=\frac{1}{2}$.

Step5: Calculate probability of yellow given odd

$P(Y|O)=\frac{P(O\cap Y)}{P(O)}=\frac{\frac{1}{10}}{\frac{1}{2}}=\frac{1}{5}$.

Step6: Check independence

Two events $A$ and $B$ are independent if $P(A|B)=P(A)$ and $P(B|A)=P(B)$. Here $P(Y|O) = P(Y)=\frac{1}{5}$ and $P(O|Y)=\frac{P(O\cap Y)}{P(Y)}=\frac{\frac{1}{10}}{\frac{1}{5}}=\frac{1}{2}=P(O)$.

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