# Simple probability proof using conditional probabilities

• February 12th 2010, 03:08 PM
sirellwood
Simple probability proof using conditional probabilities
Im terrible at proofs....

Use the definition of conditional probabilities to prove that any events A, B, C, D, E and F,

P(A $\cap$B $\cap$C $\cap$D $\cap$E $\cap$F) = P(A $\cap$B $\cap$C $\cap$D $\mid$E $\cap$F)P(E $\cap$F)

and P(A $\cap$B $\cap$C $\cap$D $\cap$E $\cap$F) = P(A $\cap$B $\mid$C $\cap$D $\cap$E $\cap$F)P(C $\cap$D $\cap$E $\mid$F)P(F)

Also, can anyone form a similar identity?
• February 12th 2010, 04:09 PM
johanS
they all come from the definition of conditional probability:
$P(A | B) = \frac{P(A \cap B)}{P(B)} \Rightarrow P(A \cap B) = P(A|B)P(B)$
and the associative law of intersection of sets:
$(A \cap B) \cap C = A \cap ( B \cap C )$
The last law allaws you to write the left or right side of the equality as $A \cap B \cap C$