Three events E1 , E2 and E3

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M.Mashal

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Three events E1 , E2 and E3, defined on the same sample space Ω, have probabilities P(E1) = P(E2) = P(E3) = 1/4. Let E0 be the event that one or more of the events E1,E2,E3 occurs.


1. Find P(E0) when:
(a) Events E1,E2 and E3 are disjoint.
(b) Events E1,E2 and E3 are independent.
(c) Events E1,E2 and E3 are, in fact, three names for the same event.


2. Find the maximum value P(E0) can assume when:
(a) Nothing is known about the independence or disjointness of E1,E2 and E3.
(b) Events E1,E2 and E3 are pairwise independent, that is, P(Ei ∩Ej) = P(Ei)P(Ej);1 ≤ i ≠ j ≤ 3, but nothing is known about P(E1∩E2∩E3).
 
M.Mashal said:
Three events E1 , E2 and E3, defined on the same sample space Ω, have probabilities P(E1) = P(E2) = P(E3) = 1/4. Let E0 be the event that one or more of the events E1,E2,E3 occurs.

1. Find P(E0) when:
(a) Events E1,E2 and E3 are disjoint.
(b) Events E1,E2 and E3 are independent.
(c) Events E1,E2 and E3 are, in fact, three names for the same event.

2. Find the maximum value P(E0) can assume when:
(a) Nothing is known about the independence or disjointness of E1,E2 and E3.
(b) Events E1,E2 and E3 are pairwise independent, that is, P(Ei ∩Ej) = P(Ei)P(Ej);1 ≤ i ≠ j ≤ 3, but nothing is known about P(E1∩E2∩E3).
Click to expand...
What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you! ;)
 
M.Mashal said:
Three events E1 , E2 and E3, defined on the same sample space Ω, have probabilities P(E1) = P(E2) = P(E3) = 1/4. Let E0 be the event that one or more of the events E1,E2,E3 occurs.
1. Find P(E0) when:
(a) Events E1,E2 and E3 are disjoint.
(b) Events E1,E2 and E3 are independent.
(c) Events E1,E2 and E3 are, in fact, three names for the same event.


2. Find the maximum value P(E0) can assume when:
(a) Nothing is known about the independence or disjointness of E1,E2 and E3.
(b) Events E1,E2 and E3 are pairwise independent, that is, P(Ei ∩Ej) = P(Ei)P(Ej);1 ≤ i ≠ j ≤ 3, but nothing is known about P(E1∩E2∩E3).
Click to expand...
Here are two basic definitions.
Mutually disjoint and Independent events.
Answers #1
A) \(\displaystyle \frac{3}{4}\)_________ B) \(\displaystyle \frac{3}{4}-\binom{3}{2}\frac{1}{16}+\frac{1}{64}\)_________ C)\(\displaystyle \frac{1}{4}\)

#2 I confess to not knowing what this question really means.
 
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