Question

(a) experiment 1: a coin is flipped. then a number cube with sides labeled 1 through 6 is rolled.
event a: the coin toss is heads.
event b: the number cube roll is a 3.
determine whether events a and b are independent or dependent.
independent dependent
(b) experiment 2: a paper clip is randomly selected from a container with green clips and white clips and returned to the
container. the paper clips are mixed. then another random selection is made.
event a: the first selection is a white paper clip.
event b: the second selection is a green paper clip.
determine whether events a and b are independent or dependent.
independent dependent
(c) experiment 3: a box contains assorted chocolates. some chocolates contain only nuts and the others contain only
cherries. a chocolate is randomly selected and eaten. then another random selection is made from the remaining
chocolates.
event a: the first selection contains nuts
event b: the second selection contains cherries.
determine whether events a and b are independent or dependent.
independent dependent

Explanation:

• (a) Flipping a coin and rolling a number cube: The outcome of the coin - toss (Event A) has no impact on the outcome of the number - cube roll (Event B). The probability of getting heads on the coin is \(P(A)=\frac{1}{2}\), and the probability of rolling a 3 on the number cube is \(P(B)=\frac{1}{6}\). The probability of both events \(A\) and \(B\) occurring is \(P(A\cap B)=P(A)\times P(B)=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}\).
• (b) Selecting a paper - clip with replacement: When the first paper - clip is selected and then returned to the container, the composition of the container (number of green and white clips) remains the same for the second selection. So, the outcome of the first selection (Event A) does not affect the outcome of the second selection (Event B).
• (c) Selecting chocolates without replacement: When the first chocolate is eaten (not replaced), the total number of chocolates and the number of chocolates of each type (nut - containing and cherry - containing) change for the second selection. So, the outcome of the first selection (Event A) affects the probability of the second selection (Event B).

Answer:

• (a) Independent
• (b) Independent
• (c) Dependent