Question

alberta diploma question
alan places five white marbles and five black marbles into a bag. he then performs the two experiments described below to select two marbles from the bag.
first experiment
one marble is selected from the bag and replaced before a second marble is selected.
second experiment
one marble is selected from the bag and not replaced before a second marble is selected.
the following two events are the same for each experiment:
event x: the first marble selected is black.
event y: the second marble selected is white.
in the first experiment, event x and event y are select an answer , and in the second experiment, event x and event y are select an answer
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dependent
independent

Explanation:

Step1: Define independent and dependent events

Independent events: The occurrence of one event does not affect the probability of the other event. Dependent events: The occurrence of one event affects the probability of the other event.

Step2: Analyze the first experiment

In the first experiment, the marble is replaced after the first - draw. The probability of event X (first marble is black) is $P(X)=\frac{5}{10}=\frac{1}{2}$. The probability of event Y (second marble is white) is $P(Y)=\frac{5}{10}=\frac{1}{2}$. The probability of Y does not depend on whether X has occurred or not since the marble is replaced. So, event X and event Y are independent.

Step3: Analyze the second experiment

In the second experiment, the marble is not replaced after the first - draw. If event X (first marble is black) occurs, then there are 9 marbles left in the bag and 5 white marbles. So the probability of event Y given that X has occurred is $P(Y|X)=\frac{5}{9}$. If X does not occur (first marble is white), then the probability of Y is $P(Y|\text{not }X)=\frac{4}{9}$. The probability of Y depends on whether X has occurred or not. So, event X and event Y are dependent.

Answer:

In the first experiment, Event X and Event Y are independent.
In the second experiment, Event X and Event Y are dependent.