9.7 Probability

9.7 Probability 

An experiment is an observed activity in which the result is uncertain, such as tossing a coin or rolling a dice.

Each possible observation is called an outcome.

The set of possible outcomes is called the sample space.
example for flipping one coin
S = {H, T}example for flipping two coins
S = {HH, HT, TH, TT}

Any subset of a sample set is called an event.

If the event (E) has an equal likelihood of outcomes you can find its probability. To calculate the probability of an event, find the number of outcomes n(E) and divide that by the sample space n(S). 

The probability of event E is:

Since the sample space is greater than or equal to the outcomes its true that 0 ≤ P(E) ≤ 1.

If P(E) = 0 it is called an impossible event. 

If P(E) = 1 it is called a certain event. 

Independent Events

Events are independent if the occurrence of one has no effect on the occurrence of another. (It occurs with replacement so for example if you have a bag of balls you must replace the ball after each draw in order to make sure the probability is not changed for the next draw.)

If A and B are independent events, the probability that both A and B will occur is:

P(A and B) = P(A) • P(B)

Mutually Exclusive 

Two events A and B are mutually exclusive if A and B have no outcomes in common or denoted as,

P(A ∩ B) = Ø 

In general, if A and B are events in the same sample space,  the probability of A or B occurring is: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) 

However if the event is mutually exclusive, then 

P(A ∪ B) = P(A) + P(B) 

(this is because the P(A ∩ B) would just be zero) 

Collectively Exhaustive

An event is collectively exhaustive if P(A ∪ B) = U

(where U = the universal set)

Collectively exhaustive is the complement of mutually exclusive.

The Complement of an Event

The complement of an event A is the collection of all of the outcomes in the sample space that are not in A. The complement does not have a standard denotation but some common ones include:

The probability of Ā is:

P(Ā) = 1 - P(A)