Question

P(EF') = P(E) - P(EF)
Is it true or false? if its false how?

Answer 1

is this probability?

Answer 2

yes

Answer 3

\[P(E\cap F)=P(E)-F(E\cap F^c)\]?

Answer 4

yeah so is it true?

Answer 5

yeah it is true

Answer 6

no not F(EF') its P(EF')

Answer 7

first of all by previous exercise we know that
\[E=(E\cap F)\cup (E\cap F^c)\]

Answer 8

yeah that was as typo

Answer 9

oh ok lol

Answer 10

and since
\[E\cap F\] and \[E\cap F^c\] are disjoint, the probability of their union is the sum of their probabilities, that is
\[P((E\cap F)\cup (E\cap F^c))=P(E\cap F) +P (E\cap F^c)\]

Answer 11

therefore since the sets are the same, you have
\[P(E)=P(E\cap F) +P (E\cap F^c)\]

Answer 12

ohh thank you

Answer 13

if you think about what this says in english it is obvious. you are interested in the probability of E
so you know you are in the set E. now if you are in E either you are in F or you are not in F
those are the logical possibilities. so
\[E=(E\cap F)\cup (E\cap F^c)\]