Hello, Amy-Marie!
What is the formula of a dependent probability for P(A and B),
if it's different from the formula for a independent formula for P(A and B)?
Independent ---> P(A and B) = P(A) * P(B)
Dependent --->P(A and B)= ???
In symbols: \(\displaystyle \,P(A\,\cap\,B)\;=\;P(A)\cdot P(B|A)\)
In English: The probability that A happens . . . times . . .
\(\displaystyle \;\;\)the probabiity that B happens, given that A has already happened.
Example: Two cards are drawn from a deck of cards without replacement.
. . . . . . . . What is the probabiity that both are Hearts?
A = "first card is a Heart"
B = "second card is a Heart"
Since there are 52 cards of which 13 are Hearts: \(\displaystyle \,P(A)\,=\,\frac{13}{52}\)
What is \(\displaystyle P(B)\) . . . prob. that the second is a Heart?
We might argue that
it depends!
\(\displaystyle \;\;\)It depends on whether the first card was a Heart ... or not.
But the formula says that we can
assume that the first card <u>was</u> a Heart.
\(\displaystyle \;\;\)So there are 51 cards left and 12 of them are Hearts: \(\displaystyle \;P(B)\,=\,\frac{12}{51}\)
Therefore: \(\displaystyle \,P(A\,\cap\,B)\:=\:\left(\frac{13}{52}\right)\left(\frac{12}{51}\right)\:=\:\frac{1}{17}\)
