# Probability Question

KevinItIs
I have a GRE question of which there are two possible solutions, both of them seem correct to me, but I can't decide which line of reasoning is right. So Its my humble request to the people here to help me decide which one is correct and WHY.
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Given the probability of happening an event A is 0.80 and event B is 0.60.
Col A: The Probability of happening event A or B
Col B: 0.92

a)Col A is Greater.
b)Col B is Greater.
c)Both Col's are equal.
d)Answer cannot be determined from the information given.
.......................

Let P(A)= Probability of A happening = 0.80 ; P(B)= Probability of B happening = 0.60;
P(A')= Probability of A NOT happening = 0.20 ; P(B')= Probability of B NOT happening = 0.40;

Solution 1: P(A)P(B') + P(A')P(B) + P(A')P(B') = 0.8*0.4+0.2*0.6+0.8*0.6 = 0.32+0.12+0.48
=0.92. So on this basis, the answer is "C".

Solution 2: Prob that either A or B happens is 1 minus probability that neither happens. So
Prob = 1 - (0.20*0.40) = 0.20. So on this basis, the answer is "B".
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Please Clarify which line of reasoning and hence the solution is correct.

Hi Kevin. The correct answer is "d)". It cannot be determined from the data given.

Note that if we were to assume that that two events were independent then "c)" would indeed be correct. As it stands however all that we can really say about P(A or B) is that it's between 0.8 and 1 inclusive.

KevinItIs
Hi Kevin. The correct answer is "d)". It cannot be determined from the data given.

Note that if we were to assume that that two events were independent then "c)" would indeed be correct. As it stands however all that we can really say about P(A or B) is that it's between 0.8 and 1 inclusive.

Can you please explain why the line of reasoning that points to choice "b" is incorrect? Seems to me there isn't anything wrong with it. But of course I could be wrong, all I want to know is to why the second line of reasoning can't be utilized? It doesn't assume that the events are independent and hence satisfies the conditions in the question...

Can you please explain why the line of reasoning that points to choice "b" is incorrect? Seems to me there isn't anything wrong with it. But of course I could be wrong, all I want to know is to why the second line of reasoning can't be utilized? It doesn't assume that the events are independent and hence satisfies the conditions in the question...

Ok the second attempt is just plain sloppy arithmetic, nothing more. 0.4 * 0.2 = 0.08 (not 0.8 - oh the shame).

In either case however you are assuming that the events are independent. You do this implicitly when you multiply probabilities.

In general Prob(A and B) is not equal to P(A) times P(B). That's why "d" is the only correct answer.

Homework Helper
Given the probability of happening an event A is 0.80 and event B is 0.60.
Col A: The Probability of happening event A or B
Col B: 0.92

All we can determine about Col A is that the probability of A or B happening is between max(P(A), P(B)) = 0.80 and min(1, P(A) + P(B)) = 1. If Col B were, say, 0.75 you could confidently say that Col A was greater. But since it's inside the range, it depends on the distribution of A and B.

KevinItIs
Thank You both, CRGreathouse and Uart for your insights. Okay, the answer is D if that's the question. But let's change the question slightly. Let us assume both A and b are independent events. What follows in that case? Solution 1? Or 2?

What you call sloppy mathematics [except the error in calculation, sorry bout that, and sorry for the misinterpretation if miscalculation is what you called sloppy and not the approach] is actually a formula I double checked from an old high school mathematics book. The author states in this book that in cases where "ATLEAST ONE" of the independent events is to occur, the way to calculate the probability is [ 1 - (P(A')P(B')) ] where A and B are independent events. I guess the clause "A or B" must translate to "Atleast A or B". If not, the first line of reasoning should just assume P(A)P(B')+ P(B)P(A') and exclude the term P(A)P(B). I overlooked the "error" in my first post where its mentioned as "P(A')P(B')" [but its not my solution its what one of my friends thought.]

So? What should be the answer if we assume A and B as independent.

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That said, I am requesting the answers to these questions most humbly and the doubts I have are just some misunderstandings in search of knowledge. I feel that if I can clarify some of these basics, the questions and doubts just resolve themselves out.

Thank you again for the help. Both of You.

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KevinItIs
All we can determine about Col A is that the probability of A or B happening is between max(P(A), P(B)) = 0.80 and min(1, P(A) + P(B)) = 1. If Col B were, say, 0.75 you could confidently say that Col A was greater. But since it's inside the range, it depends on the distribution of A and B.

Sorry but the max probability is 0.80 and the minimum is 1 !? Or am I misinterpreting something?

Homework Helper
Sorry but the max probability is 0.80 and the minimum is 1 !? Or am I misinterpreting something?

Yes, you're misinterpreting it entirely. :tongue2:

The range of the probabilities is from
the greater of P(A) and P(B)
and
the lesser of 1 and P(A) + P(B)
.

Thank You both, CRGreathouse and Uart for your insights. Okay, the answer is D if that's the question. But let's change the question slightly. Let us assume both A and b are independent events. What follows in that case? Solution 1? Or 2?

What you call sloppy mathematics [except the error in calculation, sorry bout that, and sorry for the misinterpretation if miscalculation is what you called sloppy and not the approach] is actually a formula I double checked from an old high school mathematics book. The author states in this book that in cases where "ATLEAST ONE" of the independent events is to occur, the way to calculate the probability is [ 1 - (P(A')P(B')) ] where A and B are independent events. I guess the clause "A or B" must translate to "Atleast A or B". If not,
the first line of reasoning should just assume P(A)P(B')+ P(B)P(A') and exclude the term P(A)P(B). I overlooked the "error" in my first post where its mentioned as "P(A')P(B')" [but its not my solution its what one of my friends thought.]

So? What should be the answer if we assume A and B as independent.

.......................
That said, I am requesting the answers to these questions most humbly and the doubts I have are just some misunderstandings in search of knowledge. I feel that if I can clarify some of these basics, the questions and doubts just resolve themselves out.

Thank you again for the help. Both of You.
Hi Kevin. When you fix the arithmetic error both of your initial approaches give exactly the same result (of "c" = 0.92).

Both of the approaches you used were correct.

Your first method was basically a decomposition into mutually exclusive components : P(A or B) = P(~A B) + P(A ~B) + P(A B).

Your second approach is basically De-Morgan's Theorem : (A or B) = ~(~A AND ~B).

A third method (my preferred) would have been to use : P(A or B) = P(A) + P(B) - P(A B). This last method uses the fact that P(A) includes the probability that A alone occurs plus the probability that both A and B occur. Similarly P(B) also includes the probability that both A and B occur, so that P(A) + P(B) actually counts P(A B) twice hence the need to subtract it once. This is the most common method for this type of problem.

As you can easily check, if you make no arithmetic errors then all three of the above approaches give the answer of 0.92 if the events are independent.