set question from Goldberg's book

**ptremblay** · 05-13-2012 05:22 PM

I am working my way through Goldberg's Probability: An Introduction and am stuck on problem 4.13, chapter 2. The problem states: "Let E and F be any two events. Suppose the numbers P(E), P(F), and $p(E \cap F)$ are known. Find numbers for these terms for the following probabilities:... $P(E' \cup F)$ ."

I can only get as far as:

$P(E' \cap F) = P(E') + P(F) - P(E' \cap F) = 1 - P(E) + P(F) - P(E' \cap P(F))$

I have no idea how to convert the right most part, the intersection of the compliment of E and F into the knowns. I have tried DeMorgan's laws with no help.

Oh, the answer in the back of the book is:

$1-P(E) + P(E \cap F)$

**Dason** · 05-13-2012 05:33 PM

Originally Posted by ptremblay

I can only get as far as:

$P(E' \cap F) = P(E') + P(F) - P(E' \cap F) = 1 - P(E) + P(F) - P(E' \cap P(F))$

I'm assuming your left hand side is supposed to be $P(E' \cup F)$

I like your first step. The second step I like one part of it (writing P(E') = 1-P(E)) but I don't like the very last part (how can an event be intersected with a probability?)

One hint I'll give is that hopefully you can recall that
$F = (F \cap E) \cup (F \cap E')$ and that $F\cap E$ and $F \cap E'$ are disjoint. So turn that into a probability statement and see if you can use that to finish up what you want to show.

**ptremblay** · 05-13-2012 08:50 PM

Woops! I should double check my work. Meant to write:

$P(E' \cup F) = P(E') + P(F) - P(E' \cap F) = 1 - P(E) + P(F) - P(E' \cap F)$

The text book did not give either of your identities. So:

$1 - P(E) + P(F) - P(E' \cap F) = 1 - P(E) + P ((F \cap E) \cup (F \cap E')) - P(E' \cap F)$

$= 1 - P(E) + P(F \cap E) + P(F \cap E') - P((F\cap E) \cap P(F \cap E')) - P(E' \cap F)$

$= 1 - P(E) + P(F \cap E) - P(E' \cap F) + P(E' \cap F) - P((F\cap E) \cap P(F \cap E'))$

$= 1 - P(E) + P(F \cap E) - 0 - 0$

$= 1 - P(E) + P(F \cap E)$

I never would have gotten this without your help, since, as I stated, the book did not give the identities. (I hate when textbooks give you problems they haven't readied you to solve.)

Thanks!

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