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Probability

Some basic probability concepts.

Probability Logic

Name Calculation Venn Diagram Description
A Not/A Complement 1 - P(A) = P(A’) A Not  
B Not/B Complement 1 - P(B) = P(B’) B Not  
And/Intersection P(A) * P(B) = P(A ∩ B) And Probability of A and B
Or/Union P(A) + P(B) - (P(A) * P(B)) = P(A ∪ B) Or Probability of A or B or both
Xor P(A) + P(B) - (2 * P(A) * P(B)) = P(A Δ B) xor Probability of A or B, but not both
Neither nor 1 - P(A ∪ B) = P((A ∪ B)’) nn Probability fo neither A nor B
A Not B P(A) * P(B’) anotb Probability of A, but not B
B Not A P(B) * P(A’) bnota Probability of B, but not A

Law of Large Numbers

A the number of outcomes for a probabilistic event goes to infinity the outcomes get closer and closer to the true probability of that event.

Ex: If you flip a coin an infinite amount of times, 50% of the outcomes will be heads and 50% tails.

Conditional Probability

If P(A) = P(A|B) then B doesn’t effect the probability of A.

If P(B) = P(B|A) then A doesn’t effect the probability of B.

Name Calculation Description
Conditional $\frac{P(A ∩ B)}{P(B)}$ = P(A|B) Probability of A given that B happened.

Bayes Theorem

Bayes theorem is a formula that is used to help you update your believes based upon new evidence. New evidence shouldn’t determine your believes, but update your already existing believes.

Symbol Description
P(A) Probability of A being true before the evidence.
P(B) Probability of B being true before the evidence.
P(E|A) Probability of the event happening given A is true
P(E|B) Probability of the event happening given B is true
P(A|E) Probability of A given the event
P(B|E) Probability of B given the event

P(A) = P(B’) and P(B) = P(A’)

P(A|E) = P(B|E’) and P(B|E) = P(A|E’)

$P(A|E) = \frac{P(A) * P(E|A)}{P(A) * P(E|A) + P(B) * P(E|B)}$

Example:

There is a meek and tidy person. Are they more likely to be a librarian or a farmer?

A = Librarian B = Farmer E = They are a meek and tidy person

Symbol Description Estimated value Estimated Description
P(A) Probability of them being a librarian 1/21 There are 21 times more farmers than librarians
P(B) Probability of them being a farmer 20/21  
P(E|A) Probability of someone who is meek and tidy being a librarian 4/10 40% of librarians are meek and tidy
P(E|B) Probability of someone who is meek and tidy being a farmer 1/10 10% of farmers are meek and tidy

What is P(A|E)? What’s the probability of the meek and tidy person being a librarian?

$P(A|E) = \frac{.04761 * .4}{.04761 * .4 + .95239 * .1} = .16667$

or

$P(A|E) = \frac{1 * .4}{1 * .4 + (21 - 1) * .1} = .16667$

Even if librarians are 4 times more likely to be meek and tidy, it doesn’t mean that the person is more likely a librarian.