Solution Found!
(a) Prove that if E and F arc mutually exclusive, then (b)
Chapter 3, Problem 5TE(choose chapter or problem)
(a) Prove that if E and F are mutually exclusive, then
\(P(E \mid E \cup F)=\frac{P(E)}{P(E)+P(F)}\)
(a) Prove that if \(E_i,i \ge 1\) and F are mutually exclusive, then
\(P\left(E j \mid \cup_{i=1}^{\infty} E_{i}\right)=\frac{P\left(E_{j}\right)}{\sum_{i=1}^{\infty} P\left(E_{i}\right)}\)
Questions & Answers
QUESTION:
(a) Prove that if E and F are mutually exclusive, then
\(P(E \mid E \cup F)=\frac{P(E)}{P(E)+P(F)}\)
(a) Prove that if \(E_i,i \ge 1\) and F are mutually exclusive, then
\(P\left(E j \mid \cup_{i=1}^{\infty} E_{i}\right)=\frac{P\left(E_{j}\right)}{\sum_{i=1}^{\infty} P\left(E_{i}\right)}\)
ANSWER:Step 1 of 2
Our goal is
a). We need to prove P=
.
b). We need to prove P=
.
Given E and F are mutually exclusive.
a). Now we are assuming at least one of E or F has positive probability.
Then with probability 1, so one of them will occur.
The probability of E given (EF) is
P=
P=
Here 0
.
Then,
P=
Hence we proved.