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:

To determine if events "odd" (let's call this event \( A \)) and "yellow" (event \( B \)) are independent, we use the definition of independent events: \( P(A) = P(A|B) \) (the probability of \( A \) is equal to the probability of \( A \) given \( B \)).

Step 1: Find total number of slips and odd numbers

• Total slips: \( n = 20 \)
• Number of odd numbers (1 - 20): There are 10 odd numbers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19), so \( P(A) = \frac{10}{20} = \frac{1}{2} \).

Step 2: Find number of yellow slips and odd numbers on yellow slips

• Yellow slips: \( n(B) = 4 \)
• Odd numbers on white slips: 8, so odd numbers on yellow slips: Total odd (10) - odd on white (8) = 2. Thus, number of odd numbers on yellow slips is 2.
• Probability of odd given yellow: \( P(A|B) = \frac{\text{Odd on yellow}}{\text{Yellow slips}} = \frac{2}{4} = \frac{1}{2} \).

Step 3: Compare \( P(A) \) and \( P(A|B) \)

Since \( P(A) = \frac{1}{2} \) and \( P(A|B) = \frac{1}{2} \), \( P(A) = P(A|B) \). This means the events are independent because the probability of choosing an odd number is equal to the probability of choosing an odd number given the slip is yellow.

Answer:

C. 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