Hi
My question is
State and prove the theorem of total probability ?
I tried looking for this proof in my textbook but i can't seem to find it anywhere
any help is appreichated
thanks
Are you looking for the proof of this theorem? i wasnt sure
I am not sure if i can post external links on this website...
Check out
Baye's Theorem explained with proof | TutorVista
it has a proof of this theorem
Hello,
Law of total probability :
For a partition $\displaystyle (A_n)_{n\geq 1}$ of $\displaystyle \Omega$, the probability space, for any event B (and for higher levels, considering the measured space $\displaystyle (\Omega,\mathcal{A},\mathbb{P})$, for any $\displaystyle B\in\mathcal{A}$...), we have :
$\displaystyle \mathbb{P}(B)=\sum_{n=1}^\infty \mathbb{P}(B|A_n)\mathbb{P}(A_n)$
and the proof.... :
$\displaystyle \bigcup_{n=1}^\infty A_n=\Omega$
So $\displaystyle B=B\cap \Omega=B\cap \bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty \{B\cap A_n\}$
But the last union is a disjoint union : $\displaystyle \forall i\neq j~,~ \{B\cap A_i\}\cap \{B\cap A_j\}=\emptyset$ (very easy to prove)
Thus the probability of the union is just the sum of the probabilities :
$\displaystyle \mathbb{P}(B)=\sum_{n=1}^\infty \mathbb{P}(B\cap A_n)$
Then just use the formula $\displaystyle \mathbb{P}(A\cap B)=\mathbb{P}(B|A)\mathbb{P}(A)$