Probability

Permutation formula for total combinations

n!/(n-r)!

n → total items

r → members in one group

Stars and bars technique

In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems.

Transitive dice

Transitive dice are a special type of dice where the faces are labeled with numbers such that when you roll them and rearrange the order, you can obtain any other possible combination. In other words, if you have two transitive dice, and you roll both of them, then re-arrange the numbers on one of them, you can obtain any possible result that you could get from rolling two regular dice.

All probabilities are…

-Conditional

P( A | B) =

| A n B | / | B |

P ( A n B ) =

P(A | B ) * P(B)

Tests for independent events

P(A) = P(A | B)

P(A n B) = P(A)P(B)

What is the different between independent events and disjoint events?

-Independent → P (A n B) = P(A)P(A)

-Disjoint → P(A n B) = 0 no overlap

• Independent events: Events are independent if the occurrence of one event doesn't affect the probability of the other event happening. They're like separate occurrences that don't influence each other.

• Disjoint events: Disjoint events, also known as mutually exclusive events, are events that cannot happen at the same time. If one event occurs, the other event cannot occur simultaneously. They have no outcomes in common.

P( A | B) ≠

P( B |A )

Law of total probability

P(A)=∑i=1n​P(A∩Bi​)

Special case of law of total probability

P(A) = P(A | B)P(B) + P(A | B) (1-P(B))