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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   65% (hard)

Question Stats: 45% (01:11) correct 55% (01:06) wrong based on 200 sessions

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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) Probability that $$A$$ will happen is 0.25

(2) Probability that $$B$$ will NOT happen is 0.71

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Math Expert V
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Official Solution:

Notice that since events $$A$$ and $$B$$ are independent, then the probability that both occur, equals to the product of their individual probability, so $$P(A \text { and } B)=P(A)*P(B)$$. Also notice that $$0 \le P(A) \le 1$$ and $$0 \le P(B) \le1$$.

(1) Probability that $$A$$ will happen is 0.25. Now, since $$P(A)=0.25$$, then $$P(A \text { and } B)=P(A)*P(B) \le 0.25 \lt 0.3$$. 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 happen is 0.71. The same here: since $$P(B)=1-0.71=0.29$$, then $$P(A \text { and } B)=P(A)*P(B) \le 0.29 \lt 0.3$$. Sufficient.

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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.
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coolparthi wrote:
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.

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)
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JJo wrote:
coolparthi wrote:
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.

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)

Thanks. Did not realize that!!
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I think this question is good and helpful.
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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.
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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.
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I think this is a high-quality question and I agree with explanation.
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I think this the explanation isn't clear enough, please elaborate. and can you please provide a link from where i can understand probability and combinatorics
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nishantdoshi wrote:
I think this the explanation isn't clear enough, please elaborate. and can you please provide a link from where i can understand probability and combinatorics

Theory on probability problems

Data Sufficiency Questions on Probability
Problem Solving Questions on Probability

Tough Probability Questions

Theory on Combinations

DS questions on Combinations
PS questions on Combinations
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The question reads that P(A and B)>0.3, but the explanation read that P(A and B) < 0.3.

What am I missing?
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gentlestrength wrote:
The question reads that P(A and B)>0.3, but the explanation read that P(A and B) < 0.3.

What am I missing?

Not true. The question ASKS is the probability of the event greater than 0.3. From both statements we get NO answer to the question, the probability of that even is less than 0.3.
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I think this is a high-quality question and I agree with explanation.
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Great Question!!
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I think this is a high-quality question and I agree with explanation.
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Can we solve this with the help of venn diagram?
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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
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can someone please explain this concept once that probability that both will occur is less than the probability that individuals events will occur ??
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Since joint probability of events implies that both event must occur hence the joint probability cannot be more than the individual probabilities of either of the events occurring Re: M23-02   [#permalink] 06 Dec 2018, 23:35

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