CMSC250
Probability
Probability
Conditional Probability
Recall: $P(A\vert B) = \frac{P(A \cap B)}{P(B)}$
$P(A)$ given that $B$ already occured
Roll 2 distinct dice. Probability that the sum is prime?
Probability that the sum is prime, given the first dice is a 1?
Recall: $P(A\vert B) = \frac{P(A \cap B)}{P(B)}$
$P(A\vert B)(P(B)) = P(A \cap B)$
Suppose I have 52 cards. Each card has a number (1-13) 4 of each number
Draw 2, without replacement: probability I get first an even number and then a 7 or 11?
Consider: for events $A,B$, and outcome in A is either in B or it isn't.
That is $A \cap B$ and $A \cap B^c$ are disjoint
$P(A) = P(A \cap B) + P(A \cap B^c)$
$P(A) = P(A\vert B)P(B) + P(A\vert B^c)P(B^c)$
$P(A) = P(A\vert B)P(B) + P(A\vert B^c)P(B^c)$
Suppose a bag has 5 green and 7 purple balls. You take out one and then another. Probability second is green?
$A = $ The second draw is green
$B = $ The first draw is purple
Bayes Theorem
Question: Are $P(A\vert B)$ and $P(B\vert A)$ related?
$P(B\vert A) = \frac{P(A \cap B)}{P(A)}$
$P(A\cap B) = P(B\vert A)P(A)$
$P(A\vert B) = \frac{P(A\cap B)}{P(B)}$
$P(A\vert B) = \frac{P(B\vert A)P(A)}{P(B)}$
$P(A\vert B) = \frac{P(B\vert A)P(A)}{P(B)}$
Bayes Theorem
Useful for prior knowledge probabilities
$P(A\vert B) = \frac{P(B\vert A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}$
Suppose there is a medical test to see if you have a virus
Test has 95% true positive rate
Test has 2% false positive rate
3% of population have virus
Probability you have virus if you test positive?
Suppose we are doing some Natrual Langauge Processing (NLP)
A noun is followed by a verb 40% of the time
A non-noun is followed by a verb 20% of the time
10% of all words are nouns
Given a verb, probability a noun preceeded it?
Baysean Inference
$P(H\vert E) = \frac{P(E\vert H)P(H)}{P(E)}$
$H$ is the hypothesis, $E$ is the evidence
How does $H$ change given more and more evidence?
Take CMSC421 or CMSC422