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There is a 10% chance that it won't snow all winter long

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There is a 10% chance that it won't snow all winter long [#permalink]

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There is a 10% chance that it won't snow all winter long. There is a 20% chance that schools will not be closed all winter long. What is the greatest possible probability that it will snow and schools will be closed during the winter?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

I have seen several explanations on this one and I cannot understand the point of intersection I guess...Has anyone seen similar problems they saw here (specifically with given probabilities) or perhaps I am looking for a parallel (gumballs, etc) to this problem.

There is one rule on dependent probability P(A) + P(B) - P(AB) but clearly this is not the case here. The dice are independent. With colored balls, we could calculate what happens after the balls taken away (when they are not replaced) and it makes sense. With this type of problem, I do not know what to precisely think of dependent probability :( And what is up with solving probability with overlapping sets or the matrix?

And what exactly happens when we multiply 90% and 80% conceptually? I mean what does that presume and why it is not correct? Basically, a thorough breakdown of this problem would be very helpful.

Please remember that I have seen other explanations, and my questions are based on them. Probably the best of them are:
"The greatest possible probability will occur when b is dependent on a. Thus it will be their intersection set. Hence answer = 0.8"

"They're asking for the greatest possible probability. This will occur if whenever it snows, schools close. This is a causation problem. That means schools will close only if it snows, but not all the time it snows. So you don't have to multiply .9*.8, because each time .8 occurs, .9 will be in effect as well."

I do not understand exactly what they mean...Anyway, I hope you see where I am aiming.
[Reveal] Spoiler: OA

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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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hi obs,

there are 2 events
1) p(A) (probability it will snow all longer)=0.1
2) p(B)(probability the school will remain close all winter longer)= 1- 0.2 = 0.8

now if you go by rule for joint sets p(A u B) = p(A)+p(B) - p(AandB)
therefore p(AandB)= 0.1 + 0.8 - p(A u B)

now you can just think that p(AandB) could only be max when B is subset of A thereby P(A u B) would be 0.1

hence max p(AandB) would be 0.8,

I hope i made sense
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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Easily solved with a small table

Here we can infer the values in blue.

So the question asks us what is the maximum for which both C and S are satisfied. The maximum possible value for S is already 80%. For S is 90%. So the maximum for both is 80% (since anything more will no satisfy C i.e. more than 80%).

Hope it helps
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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obs23 wrote:
There is a 10% chance that it won't snow all winter long. There is a 20% chance that schools will not be closed all winter long. What is the greatest possible probability that it will snow and schools will be closed during the winter?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

I have seen several explanations on this one and I cannot understand the point of intersection I guess...Has anyone seen similar problems they saw here (specifically with given probabilities) or perhaps I am looking for a parallel (gumballs, etc) to this problem.

There is one rule on dependent probability P(A) + P(B) - P(AB) but clearly this is not the case here. The dice are independent. With colored balls, we could calculate what happens after the balls taken away (when they are not replaced) and it makes sense. With this type of problem, I do not know what to precisely think of dependent probability :( And what is up with solving probability with overlapping sets or the matrix?

And what exactly happens when we multiply 90% and 80% conceptually? I mean what does that presume and why it is not correct? Basically, a thorough breakdown of this problem would be very helpful.

Please remember that I have seen other explanations, and my questions are based on them. Probably the best of them are:
"The greatest possible probability will occur when b is dependent on a. Thus it will be their intersection set. Hence answer = 0.8"

"They're asking for the greatest possible probability. This will occur if whenever it snows, schools close. This is a causation problem. That means schools will close only if it snows, but not all the time it snows. So you don't have to multiply .9*.8, because each time .8 occurs, .9 will be in effect as well."

I do not understand exactly what they mean...Anyway, I hope you see where I am aiming.


We have p (A n B) = p(A) + p(B) - p(A u B)

p(A) is the probability that it will snow
p(B) is the probability that the schools will close

p(A n B) will be maximum when p(A u B) is minimum but p(A u B) cannot be less than the higher of the p(A) and p(B) i.e, cannot be less than 0.9.

In the above case B is totally dependent on A. i.,e schools close whenever and only when it snows. So the p(AUB) effectively becomes p(A).

The answer is p(A n B) = 0.9+0.8-0.9= 0.8

In other words the greatest probability is the probability of B.
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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Helpful guys. Appreciate your efforts. Sravna just to clarify:

We have p (A n B) = p(A) + p(B) - p(A u B)

p(A) is the probability that it will snow
p(B) is the probability that the schools will close

Quote:
p(A n B) will be maximum when p(A u B) is minimum
You mean when p(A u B) maximized, don't you?

In other words, the P(ANB) is maximized when P(AORB) is maximized and this maximization happens only when B is completely dependent on A, becoming A and P(AORB)=P(B). Seem to make sense now. Hope I won't be trapped next time.
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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New post 05 Sep 2013, 05:09
adg142000 wrote:
hi obs,

there are 2 events
1) p(A) (probability it will snow all longer)=0.1
2) p(B)(probability the school will remain close all winter longer)= 1- 0.2 = 0.8

now if you go by rule for joint sets p(A u B) = p(A)+p(B) - p(AandB)
therefore p(AandB)= 0.1 + 0.8 - p(A u B)

now you can just think that p(AandB) could only be max when B is subset of A thereby P(A u B) would be 0.1

hence max p(AandB) would be 0.8,

I hope i made sense


It looks like you came from another angle, but it seems to me there are some mix ups. P(A) - will not snow all winter long? Anyhow, was wondering if what you meant was similar to what Sravna was saying? Thanks.
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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New post 05 Sep 2013, 05:32
obs23 wrote:
adg142000 wrote:
hi obs,

there are 2 events
1) p(A) (probability it will snow all longer)=0.1
2) p(B)(probability the school will remain close all winter longer)= 1- 0.2 = 0.8

now if you go by rule for joint sets p(A u B) = p(A)+p(B) - p(AandB)
therefore p(AandB)= 0.1 + 0.8 - p(A u B)

now you can just think that p(AandB) could only be max when B is subset of A thereby P(A u B) would be 0.1

hence max p(AandB) would be 0.8,

I hope i made sense


It looks like you came from another angle, but it seems to me there are some mix ups. P(A) - will not snow all winter long? Anyhow, was wondering if what you meant was similar to what Sravna was saying? Thanks.



yeah i just missed that ,, you are right p(A) would be 0.9, blunder on my part ,,
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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obs23 wrote:
Helpful guys. Appreciate your efforts. Sravna just to clarify:

We have p (A n B) = p(A) + p(B) - p(A u B)

p(A) is the probability that it will snow
p(B) is the probability that the schools will close

Quote:
p(A n B) will be maximum when p(A u B) is minimum
You mean when p(A u B) maximized, don't you?

In other words, the P(ANB) is maximized when P(AORB) is maximized and this maximization happens only when B is completely dependent on A, becoming A and P(AORB)=P(B). Seem to make sense now. Hope I won't be trapped next time.


Hi,

I meant p(A u B) is minimized i.e., the probability either A or B occurring is minimum. This minimum is equal to the higher of p(A) and p(B) because it cannot be less than that. p(A) is the higher probability. So p(AUB) is equal to p(A).

So p(A n B) = p(A) + p(B) - p(A u B) becomes
p(A n B) = p(A) + p(B) -p(A)
p(A n B) = p(B)= 0.8
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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obs23 wrote:
There is a 10% chance that it won't snow all winter long. There is a 20% chance that schools will not be closed all winter long. What is the greatest possible probability that it will snow and schools will be closed during the winter?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

I have seen several explanations on this one and I cannot understand the point of intersection I guess...Has anyone seen similar problems they saw here (specifically with given probabilities) or perhaps I am looking for a parallel (gumballs, etc) to this problem.

There is one rule on dependent probability P(A) + P(B) - P(AB) but clearly this is not the case here. The dice are independent. With colored balls, we could calculate what happens after the balls taken away (when they are not replaced) and it makes sense. With this type of problem, I do not know what to precisely think of dependent probability :( And what is up with solving probability with overlapping sets or the matrix?

And what exactly happens when we multiply 90% and 80% conceptually? I mean what does that presume and why it is not correct? Basically, a thorough breakdown of this problem would be very helpful.

Please remember that I have seen other explanations, and my questions are based on them. Probably the best of them are:
"The greatest possible probability will occur when b is dependent on a. Thus it will be their intersection set. Hence answer = 0.8"

"They're asking for the greatest possible probability. This will occur if whenever it snows, schools close. This is a causation problem. That means schools will close only if it snows, but not all the time it snows. So you don't have to multiply .9*.8, because each time .8 occurs, .9 will be in effect as well."

I do not understand exactly what they mean...Anyway, I hope you see where I am aiming.


Ok, let's try to understand the question stem:

There is a 10% chance that it won't snow all winter long. (by the way, there is some ambiguity in this statement)
implies there is a 90% chance that it will snow all winter long i.e. 9 out of 10 winters it will snow all winter long.

There is a 20% chance that schools will not be closed all winter long.
implies there is an 80% chance that schools will be closed all winter long i.e. 8 out of 10 winters school will be closed all winter long.

What is the greatest possible probability that it will snow and schools will be closed during the winter?
Attachment:
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Ques3.jpg [ 9.22 KiB | Viewed 6316 times ]


What is the greatest possible overlap between the two? The 8 winters when the school is closed, it should snow all winter long too. That's when the overlap will be maximum. So probability is 0.8
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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Hi Karishma

While i get the logic of your answer, i always understood that probability of event A and Probability of event B occurring is P(A)*P(B). So here is applied the same and got the answer as 0.9*0.8 = 0.72.
Whats wrong with it? Please help to explain.
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There is a 10% chance that it won't snow all winter long [#permalink]

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New post 24 Jul 2015, 18:30
obs23 wrote:
There is a 10% chance that it won't snow all winter long. There is a 20% chance that schools will not be closed all winter long. What is the greatest possible probability that it will snow and schools will be closed during the winter?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

I have seen several explanations on this one and I cannot understand the point of intersection I guess...Has anyone seen similar problems they saw here (specifically with given probabilities) or perhaps I am looking for a parallel (gumballs, etc) to this problem.

There is one rule on dependent probability P(A) + P(B) - P(AB) but clearly this is not the case here. The dice are independent. With colored balls, we could calculate what happens after the balls taken away (when they are not replaced) and it makes sense. With this type of problem, I do not know what to precisely think of dependent probability :( And what is up with solving probability with overlapping sets or the matrix?

And what exactly happens when we multiply 90% and 80% conceptually? I mean what does that presume and why it is not correct? Basically, a thorough breakdown of this problem would be very helpful.

Please remember that I have seen other explanations, and my questions are based on them. Probably the best of them are:
"The greatest possible probability will occur when b is dependent on a. Thus it will be their intersection set. Hence answer = 0.8"

"They're asking for the greatest possible probability. This will occur if whenever it snows, schools close. This is a causation problem. That means schools will close only if it snows, but not all the time it snows. So you don't have to multiply .9*.8, because each time .8 occurs, .9 will be in effect as well."

I do not understand exactly what they mean...Anyway, I hope you see where I am aiming.


The greatest probability translates as "take the range of probabilities of these two events occurring under different circumstances, and choose the greatest extreme."

If event B is fully dependent on event A, then the probability is equal to the smallest probability of either event A or B, 0.8.

If event B is not fully dependent on event A, then the maximum we can know for sure is 0.72, or event A * event B.

So, the range of possibilities is 0.72 to 0.8. The greatest possibility is 0.8 or 80%.

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Basically, the probability is 0.72 when both of the events are mutually exclusive. When they are dependent you got to use the equation

P(AUB) = P(A) + P(B) - P(A intersection B)
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Could someone please explain how P(A intersection B) is calculated differently from P(A U B)? Thanks

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dina98 wrote:
Could someone please explain how P(A intersection B) is calculated differently from P(A U B)? Thanks


A intersection B is whatever is common between the two sets.

Either you will be given its value or if they are independent events, then P(A intersection B) = P(A) * P(B).

For mutually exclusive events, P(A intersection B) = 0. Mutually exclusive events are those which cannot happen at the same time such as "Getting heads on flipping a coin" and "Getting tails on flipping a coin".

When there is some dependence in the events, the maximum value the intersection can have is the lower of the two probabilities.
Say P(A) = 0.4 and P(B) = 0.7.
The maximum probability of intersection can be 0.4 because P(A) = 0.4.
The minimum value of P(A intersection B) will be 0.1 since probability cannot exceed 1 so P(A U B) is maximum 1.
1 = 0.4 + 0.7 - P(A intersection B)
P(A intersection B) = 0.1 (at least)

Actual value of P(A intersection B) will lie somewhere between 0.1 and 0.4 (inclusive)


A union B is the sum of whatever is common between them and the elements which are contained in one set only.
P(A U B) = P(A) + P(B) - P(A intersection B)
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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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Max probability of both event A & B = least value of A or B 
this is because P(A& B) = P(A)*P(B) , if the events are independent.
P(A& B) = P(A)*P(B/A) , if the events are independent.
now since the max value of probability for any event P(B) or P(B/A)= 1, hence max probability of both events = min of probability of individual events.
Here probability of snow P(A) =0.9
Probability of school closing, P(B) =0.8
Hence probability of their intersection =0.8.
Hope it is clear.
Please provide Kudos if u like the solution.

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Re: There is a 10% chance that it won't snow all winter long [#permalink]

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New post 11 Dec 2017, 23:40
Woah, that was 700 level, so cool. Somehow, it reminds me of puzzles.

There is a 10% chance that it won't snow all winter long. There is a 20% chance that schools will not be closed all winter long. What is the greatest possible probability that it will snow and schools will be closed during the winter?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

If we break it up - "There is a 10% chance that it won't snow all winter long. 90% chance it will snow

Then - "There is a 20% chance that schools will not be closed all winter long" - 80% chance it will be closed.

Question- What is the greatest possible probability that it will snow and schools will be closed during the winter? - max possibility of 90% snow and 80% closing --> which means "greatest possibility" is 80% [max chance is 80%]

Obviously, this isn't the right mathematical approach, but this is what occurred to me, and it didn't need too many calculations.
Kudos, if it was useful for you :)

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Re: There is a 10% chance that it won't snow all winter long   [#permalink] 11 Dec 2017, 23:40
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