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23 Jun 2014, 08:36
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Consider an experiment with events A, B, and C for which P(A) = 0.23 P(B) = 0.40 and P(C) = 0.85. Also, suppose that A and B are mutually exclusive and B and C are independent.
In the GMAT book they explain: Note that P(A or C) cannot be determined using the info given. But it can be determined that A and C are not mutually exclusive since P(A) + P(C) = 1.08, which is greater than 1, and therefore cannot equal P (A or C) ; from this it follows that P (A and C >= (Greater than or equal to) 0.08. One can also deduce that P(A and C) = P(C) = 0.85 since C is a subset of A ∪ C. Thus one can conclude that 0.85= (Greater than or equal to) 0.08
2. How do I know that C is a subset of A or C?
3. How do I know that A nd C is a subset of A?
4. How are these two inequalities obtained exactly ? Thus one can conclude that 0.85<= P(A or C) <= 1 and 0.08 <= P (A and C) <= 0.23
In the GMAT book they explain: Note that P(A or C) cannot be determined using the info given. But it can be determined that A and C are not mutually exclusive since P(A) + P(C) = 1.08, which is greater than 1, and therefore cannot equal P (A or C) ; from this it follows that P (A and C >= (Greater than or equal to) 0.08. One can also deduce that P(A and C) = P(C) = 0.85 since C is a subset of A ∪ C. Thus one can conclude that 0.85= (Greater than or equal to) 0.08
2. How do I know that C is a subset of A or C?
3. How do I know that A nd C is a subset of A?
4. How are these two inequalities obtained exactly ? Thus one can conclude that 0.85<= P(A or C) <= 1 and 0.08 <= P (A and C) <= 0.23
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24 Jun 2014, 23:13
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sagnik242
What are mutually exclusive events? They are those which have nothing in common e.g.
A = "It will rain tomorrow" and
B= "It will not rain tomorrow"
are mutually exclusive events. If one happens, the other cannot happen. P(A and B) = 0
Also, in any given day, either it will rain or not so P(A or B) = 1. A and B cover the entire range of possibilities.
Similarly
A = I will get an A+ on my Math test and
B = I will get a B on my Math test
are two mutually exclusive events. If you get an A+, you cannot get a B and vice versa. So P(A and B) = 0
But you could also get a C or a D so P(A or B) < 1
With mutually exclusive events, P(A or B) = P(A) + P(B) - This must be less than or equal to 1.
What about independent events? They are not connected to each other e.g.
A = I will get A+ on my Math test
B = I will get B on my English test.
It is possible that both happen. A doesn't depend on B - they are independent.
In given question, P(A) = 0.23 P(B) = 0.40 and P(C) = 0.85
A and B are mutually exclusive so P(A and B) = 0 and P (A or B) = .23+.4 = .63
Also, B and C are independent.
Since P(A) + P(C) = .23 + .85 = 1.08, we can say that A and C are not mutually exclusive. (See the highlighted part above)
So A and C must have something common.
We know the general formula P(A or C) = P(A) + P(C) - P(A and C) (similar to sets formula)
Since a probability is always <= 1, P(A or C) <= 1
P(A) + P(C) - P(A and C) <= 1
1.08 - P(A and C) <= 1
P(A and C) >= 0.08
2.
P(A or C) includes all cases where A happens or C happens or both happen. So it includes all cases where C happens (irrespective or whether A happens or not). So P(C) is a subset of P(A or C).
3.
P(A and C) includes those cases where both A and C happen. This will be a subset of all cases where A happens (irrespective or whether C happens or not). Say A happens in a total of 20 cases. A and C both will happen in less than or equal to 20 cases, right? A and C both cannot happen in more than 20 cases since A happens in 20 cases only. Out of these 20 cases, C may also happen in some cases.
25 Jun 2014, 08:30
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VeritasPrepKarishma
Great explanation, what about the 4th question I had?
