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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Explanation:

Step1: Define independent and dependent events

Independent events: Probability of one event does not affect the other. Dependent events: Probability of one event affects the other.

Step2: Analyze first experiment

In the first experiment, the marble is replaced. The probability of event X (first - black) is $\frac{5}{10}=\frac{1}{2}$. The probability of event Y (second - white) is also $\frac{5}{10}=\frac{1}{2}$ regardless of the result of the first draw. So, event X and event Y are independent.

Step3: Analyze second experiment

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

Answer:

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