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26 Oct 2021, 02:15
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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A bag contains 6 large marbles: 3 red, 2 green, and 1 yellow, and 3 small marbles: 1 red, 1 green, and 1 yellow. A single marble is selected at random from the bag. Which of the following is a pair of independent events in this scenario?
(A) Selecting a large marble and selecting a red marble.
(B) Selecting a small marble and selecting a green marble.
(C) Selecting a yellow marble and selecting a large marble.
(D) Selecting a yellow marble and selecting a red marble.
(E) Selecting a large marble and selecting a small marble.
(A) Selecting a large marble and selecting a red marble.
(B) Selecting a small marble and selecting a green marble.
(C) Selecting a yellow marble and selecting a large marble.
(D) Selecting a yellow marble and selecting a red marble.
(E) Selecting a large marble and selecting a small marble.
10 Feb 2022, 12:01
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varunn001
One definition for independence is the following: events A and B are said to be independent if P(A) = P(A | B). Here, P(A | B) is the probability of A given B, meaning the probability of A assuming event B happened. Using this definition, we can analyze each answer choice:
A) Event A is "selecting a large marble" and event B is "selecting a red marble". Thus, P(A) = 6/9 = 2/3. If we assume event B already happened, i.e. if we assume the selected marble is red, then the probability that it is a large marble is 3/4 (since there are 4 red marbles in total and 3 of them are large). Since P(A) ≠ P(A | B), events A and B are not independent.
B) Event A is "selecting a small marble" and event B is "selecting a green marble". Thus, P(A) = 3/9 = 1/3. If we assume event B already happened, i.e. if we assume the selected marble is green, then the probability that it is also small is 1/3 (since there are 3 green marbles in total and 1 of them is green). Since P(A) = P(A | B), events A and B are independent. The answer is B.
It has been suggested in earlier comments that the events in answer choice E look like they should be independent. Let's analyze answer choice E as well:
E) Event A is "selecting a large marble" and event B is "selecting a small marble". Here, P(A) = 2/3. However, if we assume that event B already happened, i.e. if the selected marble is small, then the probability that the selected marble is large is 0. In other words, P(A | B) = 0. Since P(A) is not equal to P(A | B), events A and B are not independent. In this answer choice, events A and B are mutually exclusive. It is a good idea to note that two mutually exclusive events are never independent unless one of the events have 0 probability.
General Discussion
24 Dec 2021, 08:07
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Hello,
two events are independent <-> P(A n B)=P(A)*P(B)
So let's just test that:
(A) Collecting a large marble is (6/9), collecting a red marble is in general (4/9), because there are in total 4 red marbles in the bag. That is (6/9)*(4/9). BUT: Once one draws a large marble, there are 3 out of the 6 large marbles which are red! So we would have (6/9)*(3/6), and that is not equal to our first pair of probabilities.
(B) Same procedure, but this time the prob. for drawing a small one is (3/9), drawing a green one from the bag is (3/9), so that is (3/9)^2. Now, given that I already drew a small marble, I know that 1 out of the 3 small marbles is green. So we get (3/9)*(1/3)=(3/9)^2. So (B) must be the answer
Bunuel?
No guarantees
two events are independent <-> P(A n B)=P(A)*P(B)
So let's just test that:
(A) Collecting a large marble is (6/9), collecting a red marble is in general (4/9), because there are in total 4 red marbles in the bag. That is (6/9)*(4/9). BUT: Once one draws a large marble, there are 3 out of the 6 large marbles which are red! So we would have (6/9)*(3/6), and that is not equal to our first pair of probabilities.
(B) Same procedure, but this time the prob. for drawing a small one is (3/9), drawing a green one from the bag is (3/9), so that is (3/9)^2. Now, given that I already drew a small marble, I know that 1 out of the 3 small marbles is green. So we get (3/9)*(1/3)=(3/9)^2. So (B) must be the answer
Bunuel?
No guarantees
