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

Hi, why option E is not right? because as i know the two independent events should not affect each other and option E satisfies this condition. please explain..

MMH91



Ok, first thing you have to understand is that you have to stop going by gut feeling but by rigorous definition.

Independent events are DEFINED as:

P(A n B)=P(A)*P(B).

Now, lets look at E:

Drawing a large one is 6/9, drawing a small one then is 3/8, so in total, we get (6/9)*(3/8). Now compare this to how independent events are defined, and look at the probability of drawing a large vs small marble, INDEPENDENT of what happened before:

We get 6/9 for large, and 3/9 for small (!)

This is exactly the difference: Picking a large affects the probability and (6/9)*(3/8) != (6/9)*(3/9)=P(AnB)=P(A)*P(B) <-> Two events are independent.

No guarantees

scott from TTP.

can u please provide an explanation for this

Examining B.

Independent if P(AandB)= P(A)*P(B).

P(Small not Green) =2/9
P(Green given above)=3/8
Probability of both: 6/72=1/12

P(Small and Green)= 1/9
P(Green given above)= 2/8
Probability of both:
2/72=1/36

Total probability: 1/12+1/36=
4/36= 1/9

Checking to see if this equals P(Small)*P(Green):

P(Small)= 3/9=1/3
P(Green)=3/9=1/3

P(Small)*P(Green)= 1/3*1/3=

1/9.

Since the probabilities of 1/9 are the same, the events are independent

Another approach:

If P(B/A)= P(B) then independent.

B is small, A green.

Four possibilities:

Pick small green, then green:

1/9*2/8 =1/36

Pick large green, then green:

2/9*2/8= 2/36

Pick Small not Green, then Green:

2/9*3/8= 3/36

Pick Large not Green, then Green:

4/9*3/8 = 6/36

Probability of Small given Green is then (eliminating common denominator of 36):

(1+3)/(1+3+2+6)= 4/12=

1/3.

So, if this is equal to the a priori probability of a small marble, then the events are independent:

Probability of a small marble:

3/9= 1/3

Probabilities are the same so the events are independent


Posted from my mobile device

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.
_________________

Now I'm confused.

The analysis of the other answer choices suggest that there are two draws involved.

The above however discusses large and small occurring in one draw.

I would have said P(A) = 2/3 for large ball.

P(B) = 1/3

P(A|B) is probability of a large ball on second draw given small ball drawn first.

Since there are now 8 balls left, 6 large,2 small,

Probability of large ball is now 3/4= P(A|B)

Since P(A|B)=3/4 is not equal to P(A)= 2/3, the events are not independent.

What am I missing?

Posted from my mobile device

You are right. There are two draws. Your calculations of P(A) and P(A | B) are also correct. However, the fact that P(A) is not equal to P(A | B) actually shows that A and B are not independent.
_________________

Ok thanks. Yes missed the "not"

Of course.
_________________

I found the explanation of ScottTargetTestPrep the most usefull and clear.

As he mentions, 2 events A and B are said to be independent if P(A)=P(A|B), P(A|B) being the probability of picking A within the universe of B. It is helpfull to visualize it as a matrix