Some basic probability concepts.
| Name | Calculation | Venn Diagram |
|---|---|---|
| A Not/A Complement | $1 - P(A) = P(A’)$ | |
| B Not/B Complement | $1 - P(B) = P(B’)$ | |
| And/Intersection | $P(A) * P(B) = P(A \cap B)$ | |
| Or/Union | $P(A) + P(B) - (P(A) * P(B)) = P(A \cup B)$ | |
| Xor | $P(A) + P(B) - (2 * P(A) * P(B)) = P(A \Delta B)$ | |
| Neither nor | $1 - P(A \cup B) = P((A \cup B)’)$ | |
| A Not B | $P(A) * P(B’)$ | |
| B Not A | $P(B) * P(A’)$ | |
As 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.
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 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’)
$P(A|E) = \frac{P(A) * P(E|A)}{P(A) * P(E|A) + P(B) * P(E|B)}$
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?
$P(A|E) = \frac{.04761 * .4}{.04761 * .4 + .95239 * .1} = .16667$
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.