Explanation

Since you know that the probability of event A is \(\frac{1}{2}\)and the probability of event B is \(\frac{1}{3}\) but you are not given any information about the relationship between events A and B, you can compute only the minimum possible value and the maximum possible value of the probability of the event A ∪ B. The probability of A ∪ B is least if B is a subset of A; in that case, the probability of A ∪ B is just the probability of A, or \(\frac{1}{2}\) .

The probability of A ∪ B is greatest if A and B do not intersect at all; in that case, the probability of A ∪ B is the sum of the probabilities of A and B, or \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) With no further information about A and B, the probability that A or B, or both, will occur could be any number from \(\frac{1}{2}\) to \(\frac{5}{6}\).
Of the answer choices given, only \(\frac{1}{2}\) and \(\frac{3}{4}\) are in this interval.

The correct answer consists of Choices B and C.
_________________

Need Practice? 20 Free GRE Quant Tests available for free with 20 Kudos
GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests.

This is the correct expression

P(A ∪ B) = P(A) + P(B) - P(B)P( A | B )

P(A) is probability of A happening
P(B) is probability of B happening
P( B | A ) can be said to Probability of B given that A has occured. Now look at the figure below. The area P( A | B ) is minimum when P(A) and P(B) do not overlap at all.

.
Intutive explanation of the formula:

P(A ∪ B) = P(A) + P(B) - P(B)P( A | B )

If you look at the figure to find area of P(A ∪ B) we add the area of P(A) and P(B) but if P(A) and P(B) overlap, we would add the overlapping area twice! So we sunbtract P(B)P( A | B ).

Example:

An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?


Let A = the event that the first marble is black; and let B = the event that the second marble is black.

We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

P(A) P(B|A) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133


P(B)P( A | B ) is not P(B)P(A). The amount of overlapping area cannot be found out unless specified

Please take care of this as this is the very fundamental concept of probability. Hope this helps and feel free to ask more questions.
_________________

Sandy
If you found this post useful, please let me know by pressing the Kudos Button

Try our free Online GRE Test

The answers are B and C.

It is an official question.

Regards
_________________

Need Practice? 20 Free GRE Quant Tests available for free with 20 Kudos
GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests.

Probability of event \(B\) is \(0.5\)
Probability of event \(A\) is \(0.3333\)

Given that probability of event \(B\) is greater than \(A\)
one A U B scenario could involve event \(A\) being a subset of \(B\) hence minimum possible union will be \(0.5\)
However if the two probabilities are mutually exclusive \(A + B\) would be \(0.8333\) which would be the maximum probability
as such the range of probability for A U B is \(0.5\) to \(0.8333\) inclusive
_________________

This is my response to the question and may be incorrect. Feel free to rectify any mistakes
Need Practice? 20 Free GRE Quant Tests available for free with 20 Kudos

-----------ASIDE-----------------
Key Concept #1: P(J OR K) = P(J) + P(K) - P(J and K)

Key Concept #2: P(J and K) ≤ P(J) & P(J and K) ≤ P(K)

This should make sense.
For example, P(it rains AND it snows tomorrow) must be less than or equal to P(it rains tomorrow)
---------------------------------

GIVEN:
P(A) = 1/2
P(B) = 1/3

From Key Concept #2 above, we know that P(A and B) ≤ 1/2, and we know that P(A and B) ≤ 1/3
So, the greatest possible value of P(A and B) is 1/3...
And the least possible value of P(A and B) is 0


We want to calculate P(A OR B)
P(A OR B) = P(A) + P(B) - P(A and B)
= 1/2 + 1/3 - P(A and B)

If P(A and B) = 1/3, then P(A OR B) = 1/2 + 1/3 - 1/3 = 1/2
If P(A and B) = 0, then P(A OR B) = 1/2 + 1/3 - 0 = 5/6

So, the RANGE of possible values of P(A OR B) goes from 1/2 to 5/6 (inclusive)

Check the answer choices....B and C are within the range of 1/2 to 5/6 (inclusive)

Answer: B & C

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com
Sign up for my free GRE Question of the Day emails

Don't worry about the "A ∪ B" notation. You aren't expected to know it on test day.
On test day, the question will as us to find P(A or B).

Also note that this is a VERY tricky question (to date, only 33% of students have correctly answered the question.

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com
Sign up for my free GRE Question of the Day emails

How you got \(\frac{2}{3}\)?
_________________

If you found this post useful, please let me know by pressing the Kudos Button


Rules for Posting

Got 20 Kudos? You can get Free GRE Prep Club Tests

GRE Prep Club Members of the Month:TOP 10 members of the month with highest kudos receive access to 3 months GRE Prep Club tests