a. Provide two independent events that you know the probability of. b. Explain how you know these events are independent. c. Find the probability that they both occur. a. Provide two dependent events. b. Explain how you know these events are dependent. c. Explain why these dependent events might require a different method for calculating the probability of both events occurring.

Explained below.

Step-by-step explanation:

(1)

(a)

Consider the independent events A and B as follows:

A = a coin toss results in Heads

B = a 3 is rolled on a fair dice.

P (A) = 1/2

P (B) = 1/6

(b)

Consider the experiment where a coin is tossed and a dice is rolled.

The outcome of the coin toss is not affected by the outcome of the dice rolling.

And vice-versa.

Thus, the events A and B are not dependent on each other.

(c)

Compute the probability of A and B both occurring together as follows:

; Since A and B are independent.

Thus, the probability of A and B both occurring together is 1/12.

(2)

(a)

Consider the dependent events A and B as follows:

A bag of marbles consist of 6 red, 5 blue and 4 green marbles. Two marbles are selected without replacement.

A = a green marble is selected first

B = a red marble is selected second.

P (A) = 4/15

P (B) = 6/14

(b)

Consider the experiment where two marbles are selected from the bag without replacement and their color is noted.

Since the marbled are selected without replacement, the selection of the second marble is dependent upon the first.

That is, after the first selection, the total number of marbles in the bag would be reduced by 1. Hence affecting the probability of selection of the second marble.

Thus, the events A and B are dependent on each other.

(c)

Compute the probability of A and B both occurring together as follows:

Thus, the probability of A and B both occurring together is 4/35.

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