Question

For the given pair of events A and​ B, complete parts​ (a) and​(b) below. ​

A: A marble is randomly selected from a bag containing 14 marbles consisting of 1​ red, 9 ​blue, and 4 green marbles. The selected marble is one of the green marbles. ​

B: A second marble is selected and it is the 1 red marble in the bag.

a. Determine whether events A and B are independent or dependent.​ (If two events are technically dependent but can be treated as if they are independent according to the​ 5% guideline, consider them to be​ independent.)

b. Find​ P(A and​ B), the probability that events A and B both occur.

a. Choose the correct answer below.

A. The two events are dependent because the occurrence of one affects the probability of the occurrence of the other and the​ 5% guideline can be applied in this case.

B. The two events are independent because the​ 5% guideline can be applied in this case.

C. The two events are dependent because the occurrence of one affects the probability of the occurrence of the other and the​ 5% guideline cannot be applied in this case.

D. The two events are independent because the occurrence of one does not affect the probability of the occurrence of the other.

b. The probability that events A and B both occur is ___. ​(Round to four decimal places as​ needed.)

Answer 1

Given

A marble is randomly selected from a bag containing 14 marbles consisting of 1​ red, 9 ​blue, and 4 green marbles.

A: The selected marble is one of the green (4) marbles. ​

B: A second marble is selected and it is the red (1) marble in the bag

a)

Probability for both events as follows:

P(A) = 0.2857

P(B) = 0.0769

5% guideline:

If calculations are cumbersome and if a sample size is no more than 5% of the size of the population , treat the selections as being independent.

As per guideline 14*0.05 = 0.7 ,

the probability calculation values are grater than 0.05 . so, the events are dependent.

b)

P(A and B) = P(A | B) P(B)

P(A | B) = 3 / 11

[Because , the probability that probability of drawing green marble if red marble is drawn. There is only one red marble in the bag and 4 green marbles in the bag. one red marble is already drawn in first case , there is only 3 green marbles.]

P(A | B) = 0.27

P(A B) = P(A | B) P(B)

= 0.27 * 0.0769

= 0.0208

a)

The two events are dependent because the occurrence of one affects the probability of the occurrence of the other and the​ 5% guideline can be applied in this case.(option-A)

b)

The probability that events A and B both occur is 0.0208

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