# Thread: Simple probability proof using conditional probabilities

1. ## 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$\displaystyle \cap$B$\displaystyle \cap$C$\displaystyle \cap$D$\displaystyle \cap$E$\displaystyle \cap$F) = P(A$\displaystyle \cap$B$\displaystyle \cap$C$\displaystyle \cap$D$\displaystyle \mid$E$\displaystyle \cap$F)P(E$\displaystyle \cap$F)

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

Also, can anyone form a similar identity?

2. they all come from the definition of conditional probability:
$\displaystyle P(A | B) = \frac{P(A \cap B)}{P(B)} \Rightarrow P(A \cap B) = P(A|B)P(B)$
and the associative law of intersection of sets:
$\displaystyle (A \cap B) \cap C = A \cap ( B \cap C )$
The last law allaws you to write the left or right side of the equality as $\displaystyle A \cap B \cap C$