varunn001 wrote:
scott from
TTP.
can u please provide an explanation for this
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.