Conditional Probability And Bayes Formula
How does one calculate conditional probabilites?
We shall define variables as follows:
S = Sample Space
E = Event
F = Event
From here, we can calculate a few things already..
{$P(E), P(F), P(E \cap F), P(E \cup F) …$}
What we are talking about here, however is $P(E
We will see that $P(EP(F)} \rightarrow P(E\cap F) = P(E is a very useful construction.
This will eventually lead to siplified calculations = $\frac{P(F{P(F)}.$}
Bayes’ Formula
Divide ta sample space up into overlapping parts, such that the total space can be broken into these spaces.
{$E_i \cap E_j \rightarrow \emptyset $} in general.
We partition the sample set such that {$S = \cup_{i=0}^n E_i and E_i \cap E_j = \emptyset$}
Examples of partitions are the integers split into p sets of integers modulo p + n, where n is the set.
Oh snap. so if we want to express A as unions of things in a partition, this is AOK. we hjust tae the intersection of each element of the partition.
Suppose we ahve 10 coins s.t. if that ith coin is flipped , the probability of gettinga head is i/10, where i=(1,10) (note that this is a preposterous arrangement- that is ok, it is only to motivate discussion). when one coin is seleceted randomil, and flipped, iot shows head. What is the probability that it was the 5th coin?
Sample space is {$S = \{ (i^n, H) (i^n, T) \}$}
Events: A = that the coin picked was the fifth. B = that the flipped coin showed heads
We want P(A|B). (Prob. of A given B). = \frac{P(A\cap B)}{P(B)}