# Thread: Proof of total probability theorem

1. ## Proof of total probability theorem

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

2. Originally Posted by rpatel
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

3. are bayes theorem and theorem of total probability the same thing ?

I though total probability theorem had something to do with 'paritions'.

4. Originally Posted by rpatel
are bayes theorem and theorem of total probability the same thing ?

I though total probability theorem had something to do with 'paritions'.
They are different things

in the website scroll down to Law of total probability

5. Hello,

Law of total probability :
For a partition $(A_n)_{n\geq 1}$ of $\Omega$, the probability space, for any event B (and for higher levels, considering the measured space $(\Omega,\mathcal{A},\mathbb{P})$, for any $B\in\mathcal{A}$...), we have :

$\mathbb{P}(B)=\sum_{n=1}^\infty \mathbb{P}(B|A_n)\mathbb{P}(A_n)$

and the proof.... :

$\bigcup_{n=1}^\infty A_n=\Omega$

So $B=B\cap \Omega=B\cap \bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty \{B\cap A_n\}$

But the last union is a disjoint union : $\forall i\neq j~,~ \{B\cap A_i\}\cap \{B\cap A_j\}=\emptyset$ (very easy to prove)

Thus the probability of the union is just the sum of the probabilities :

$\mathbb{P}(B)=\sum_{n=1}^\infty \mathbb{P}(B\cap A_n)$

Then just use the formula $\mathbb{P}(A\cap B)=\mathbb{P}(B|A)\mathbb{P}(A)$

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# state prove law of total probability

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