Question

(a) experiment 1: a hat contains slips of paper with the names of boys and girls in a class. a name is randomly selected and returned to the hat. the slips of paper are mixed. then another random selection is made.
event a: the first selection is a boy.
event b: the second selection is a girl.
determine whether events a and b are independent or dependent.
o independent o dependent
(b) experiment 2: a bag contains green marbles and orange marbles. a marble is randomly selected from the bag and set aside. then another random selection is made from the remaining marbles.
event a: the first selection is a green marble.
event b: the second selection is an orange marble.
determine whether events a and b are independent or dependent.
o independent o dependent

Explanation:

(a)

Step1: Analyze the first experiment

In Experiment 1, since the first - selected slip is returned to the hat, the probability of selecting a girl in the second draw (Event B) is not affected by whether a boy was selected in the first draw (Event A). The total number of slips and the number of girl - named slips remain the same for the second draw regardless of the first - draw result.

(b)

Step1: Analyze the second experiment

In Experiment 2, when the first marble (Event A) is selected and set aside, the total number of marbles for the second draw changes. If a green marble is selected first, the number of marbles left in the bag is reduced by one. This affects the probability of selecting an orange marble in the second draw (Event B). For example, if there are \(n\) marbles initially (\(g\) green and \(o\) orange, \(n = g + o\)), the probability of Event A is \(P(A)=\frac{g}{n}\). After selecting a green marble, the number of marbles is \(n - 1\), and the probability of Event B given Event A is \(P(B|A)=\frac{o}{n - 1}\), while the probability of Event B without considering Event A (if we assume no prior information) is \(P(B)=\frac{o}{n}\) (if we consider the initial state). Since \(P(B|A)
eq P(B)\) (because the sample space changes after the first draw), the events are dependent.

Answer:

(a) Independent
(b) Dependent