MHB What is the probability identity for multiple events?

[Chris L T521]

Thanks to those who participated in last week's POTW! Here's this week's problem (and the last University POTW of 2012)!

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Problem: For events $E_1$, $E_2,\ldots,\,E_n$, justify the following probability identity: \[P(E_1\cap E_2\cap\cdots\cap E_n)=P(E_1)P(E_2\mid E_1)P(E_3\mid E_2\cap E_1)\cdots P(E_n\mid E_1\cap E_2\cap\cdots \cap E_{n-1}).\]
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Hint:

Spoiler

Use the fact that $P(A\cap B) = P(A)P(B\mid A)$.

 

[Chris L T521]

This week's problem was correctly answered by CaptainBlack and Sudharaka. You can find Sudharaka's solution below:

Spoiler

We shall show this by mathematical induction. When \(n=2\) the statement is true by the definition of conditional probability. That is,\[P(E_1\cap E_2)=P(E_1)P(E_2\mid E_1)\]
Suppose that the statement is true for \(n=p\in\mathbb{Z}^{+}\). That is,
\[P(E_1\cap E_2\cap\cdots\cap E_p)=P(E_1)P(E_2\mid E_1)P(E_3\mid E_2\cap E_1)\cdots P(E_p\mid E_1\cap E_2\cap\cdots \cap E_{p-1})~~~~~~~~~~~(1)\]
Now consider \(P(E_1\cap E_2\cap\cdots\cap E_p\cap E_{p+1})\). By the definition of conditional probability we get,
\[P(E_1\cap E_2\cap\cdots\cap E_p\cap E_{p+1}) = P(E_1\cap E_2\cap\cdots\cap E_p)P(E_{p+1}\mid E_1\cap E_2\cap\cdots \cap E_{p})~~~~~~~~~~(2)\]
By (1) and (2),
\[P(E_1\cap E_2\cap\cdots\cap E_{p+1})=P(E_1)P(E_2\mid E_1)P(E_3\mid E_2\cap E_1)\cdots P(E_{p+1}\mid E_1\cap E_2\cap\cdots \cap E_{p})\]
Therefore by mathematical induction,
\[P(E_1\cap E_2\cap\cdots\cap E_n)=P(E_1)P(E_2\mid E_1)P(E_3\mid E_2\cap E_1)\cdots P(E_n\mid E_1\cap E_2\cap\cdots \cap E_{n-1})\mbox{ for }n\in\mathbb{Z}^+\]
Q.E.D.