For the given pair of events A and B, complete parts (a) and (b) below. A: A mar
ID: 3152488 • Letter: F
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 20 marbles consisting of 1 red, 15 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. 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 occurrence of one does not affect the probability of the occurrence of the other.
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 5% guideline can be applied in this case.
Explanation / Answer
a. Since the selected samples consist of 1 marble from a population of 20 marbles, the sample size is not more than 5% of the population. Therefore, the two events can be treated as independent following the 5% independence rule.
Therefore, the correct option for should be option D.
b. P(A and B)
= P(A B) = P(A)*P(B) (since the events A and B are independent)
Now, 1 marble (of any color) can be selected from 20 marbles in 20C1 = 20 ways.
And, 1 green marble can be selected from 4 green marbles in 4C1 = 4 ways.
Therefore, P(A) = 4/20 = 0.2
Similarly, P(B) = 1/20 = 0.05
Therefore, P(A B) = 0.2*0.05 = 0.01