Probability (copy)

Random Experiment

Asks for likelihood of an event happening

All outcomes are equally likely

Number of successful and total possible outcomes is countable

n(E) = number of outcomes from the event
n(S) = number of outcomes in the sample space

P(E) = n(E)/n(S)

Dependent Probability

First event changes the probability of the second event

Choosing items without replacement

The second event’s probability is conditional on the first event happening

P(A and B) = P(A) x P(B|A)

Independent Probability

Two events do not influence each other

Involves replacement or separate trials

P(A and B) = P(A) x P(B)

Overlapping Probability

Either one event happening or the other

Possible to both occur at the same time

P(A or B)=P(A)+P(B)−P(A∩B)

Mutually Exclusive Probability

Events cannot overlap

Probability of one event or the other occurring

Usually states mutually exclusive or disjoint
P(A or B) = P(A) + P(B)

Linear Permutation

Order Matters (Changing order creates a different outcome)

The problem involves arranging or ordering distinct items

There are no repeated items

P(n,r) = n!/(n−r)!

Indistinguishable Permutations

Some objects are identical

Involves arranging things

Ask for the number of unique arrangements

P(n, k) = n!/k1​!×k2​!×⋯×km​!

Circular Arrangement

Objects are arranged in a circle and rotations do not create new arrangements

The problem involves other circular setups

The order matters, but starting position does not

(n-1)!

Linear Combination

C(n,r) = n!/r!(n−r)!

Specific Probability

Identify total group size (N) and total spots to fill (R)
Determine required things
Find how many remain after the other members

C(Specific) = (Required Group C Chose Required x Remaining Pool C Remaining Spots)/(Total Group x Total Spots)

Binomial Probability

Fixed number of independent trials

n = Total number of trials

k = Number of successful trials

p = Probability of success on a single trial

1−p1 = Probability of failure on a single trial

P(X=k)=(n k​)p^k(1−p)^n−k

Mathematical Expectation

Calculating the average outcome over many trials

Making decisions based on probability

E(X)=∑xi​P(xi​)
xi​ = A possible value of XXX

P(xi)P(x_i)P(xi​) = Probability of xix_ixi​ occurring

n = Number of possible outcomes

Arithmetic Sequence Recursive Formula

Constant Differences

Explicitly Said Recursive

a1 = #

an = an-1 + d

an+1 = an + d

Arithmetic Sequence Explicit Formula

Constant Differences

Explicitly Said Explicit

an = a1 + d(n-1)

Arithmetic Series Partial Sum

Constant Differences

Explicitly Said Partial Sum

Sn = (n(a1 + an))/2

Sn = (n(2a1 + d(n-1)))/2

Geometric Sequences Recursive Formula

Constant Ratio
Explicitly Said Recursive

a1 = #
an = r x an-1
an+1 = ran

Geometric Sequences Explicit Formula

Constant Ratio
Explicitly Said Recursive

an = a1®^n-1

Geometric Series Partial Sum Formula

Constant Ratio
Explicitly Said Recursive

Sn = (a1(1 - r^n))/1-r

Geometric Series Infinite Sum

0<|r| < 1
Converges
S = a1/1-r

Restricted Linear Arrangement

Normal Linear Permutation
Objects must stay together while arranging them
(n-k+1)! x k!