M23-02

Question

If event  A  and event  B  are independent, is the probability that both event  A  and event  B  occur greater than 0.3?

(1) Probability that  A  will occur is 0.25
 
(2) Probability that  B  will NOT occur is 0.71

Bunuel · Accepted Answer

Official Solution:

If event  A  and event  B  are independent, is the probability that both event  A  and event  B  occur greater than 0.3?  
 
Notice that since events  A  and  B  are independent, then the probability that both occur, equals to the product of their individual probabilities, so  P(A 	ext { and } B)=P(A)*P(B) . Also notice that  0 \le P(A) \le 1  and  0 \le P(B) \le1 .
 
(1) Probability that  A  will occur is 0.25. 
 
Since  P(A)=0.25 , then  P(A 	ext { and } B)=P(A)*P(B) \le 0.25 \lt 0.3 . Sufficient.
 
Or consider the following: how can the probability that both Event  A  and Event  B  will happen be more than individual probability of each happening? So, the probability that both happen cannot be more than 0.25.
 
(2) Probability that B will NOT occur is 0.71. 
 
The same here: since  P(B)=1-0.71=0.29 , then  P(A 	ext { and } B)=P(A)*P(B) \le 0.29 \lt 0.3 . Sufficient.

Answer: D

shasadou · Answer

As long as the events are independent we use general case formula P(A and B)=P(A)*P(B) where ocurrence of event A does not influence occurence of event B. We also remember that probability cannot exceed 1 for any single event or combination of the events. So the question asks whether:

P(A) * P(B) > 0.3

(1) P(A) = 0.25.  Let us try to apply the formula here: under what circumstances the "and" probability of the 2 independent events can be at least equal to 0.3? Given probability of A we can build the formula and see what probability we would need for the event B for the whole formula become equal to 0.3.

0.25 * P(B) = 0.3

P(B) = 0.3/0.25 = 1.2 - hold on this cant be true, the probability of event cannot exceed 1. Hence the "and" probability of A and B just never can be 0.3 and/or greater than that. SUFFICIENT.

(2) P(not B) = 0.71  -> P(B)=0.29. Same logic - P(A) would exceed 1 which is not possible, hence SUFFICIENT to anwser the question conclusively.

coolparthi · Answer

Could you please explain this scenario.

1. Statement A provides only the probabilty of A - 0.25. 
If the probability of B is zero, then P(A) * P(B) = 0. 
If the probabilty of B is non-zero, then P(A and B) cannot be greater than 0.25

2. Statement B provides the probability of B 
If the probability of A is zero, then P(A) * P(B) = 0.

The same scenario will hold true if the probability of either one of the event is 1.

JJo · Answer

I believe the question stem is asking the probability of A&B is >0.3 ?
"If Event A and Event B are independent, is the probability that both  Event A and Event B will happen greater than 0.3 ?"
1) provides P(A)=0.25 now P(B) can be maximum 1, in this case P(A)*P(B)=0.25*1=0.25 <0.3
 min value of P(B) can be 0, in this case P(A)*P(B)=0.25*0=0 <0.3
in either case the combined probability is < 0.3, hence sufficient

2) same reasoning as 1)

Ans. D)

IonutCZ · Answer

I think this question is  good  and  helpful .

arhumsid · Answer

Beautiful question!
When i first attempted it, i thought it was sub-600 level. And i got it wrong!
After seeing the solution i realized i had all the knowledge that was required to solve this question. But, this was a classic example that shows how GMAT doesn't care about what i 'know' but the way I comprehend the given info and apply what I know.

swanidhi · Answer

I think this is a  high-quality  question and I agree with explanation.

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation.

danielarasan · Answer

Great Question!!

DmitryP · Answer

I think this is a  high-quality  question and I agree with explanation.

KB04 · Answer

It was indeed a good question.

Definition of an Independent event: Two events, A and B, are independent if the fact that event A occurs does not change the probability that event B will occur, and vice versa. Probability of event B occuring is the same regardless of whether the event A happens.

That being said, if the probability of one "independent event" is given, we can say for sure that probability of both happening will be lesser than the probability of first event because any probability is 0<x<1. That means, it's between 0 and 1. Now, if we are given that the probability of A is .25 that means it would certainly be less than the probability of A and B. To prove this point we can maximize the probability of B and take it as 1. So, .25 x 1 = .25, which is less than .30. Notice that if we reduce the probability of B the product will also reduce and cannot never be greater than .3. 
Likewise is the case for statement 2.
So, D

patto · Answer

I think this is a  high-quality  question and I agree with explanation.

amandesai17 · Answer

Good question!

carouselambra · Answer

I absolutely loved this question! Thanks Bunuel ! Definitely a fantastic example of application of common sense

rickyric395 · Answer

Since it is given A and B are independent then  P(A and B)  =  P(A)*P(B) .
Probability of any event E follows this inquality  0<=P(E)<=1

Statement 1 -----  P(A) = 0.25  , So  P(A and B)  =  0.25*P(B)  , Max value of P(B) can be 1 so  0.25*P(B)  =  0.25*1  = 0.25 i.e < 0.3.  Sufficient

Statement 2 -----  P(not B) = 1-P(B)  , given  P(not B) = 0.71  , so  P(B) = 0.29 . Using same calculations as done for statement 1. We know max value of  P(A and B)  is  1*0.29 = 0.29 .  Sufficient

Hence answer is  D

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Ankit__7182 · Answer

I think this is a high-quality question and I agree with explanation

sssdd1234 · Answer

I think this is a  high-quality  question and I agree with explanation.

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