If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.

It's used for inferential statistics 

Conduct (or observe) a procedure, and count the number of times event A actually occurs.  Based on these actual results, P(A) is estimated as follows:

P(A)=    number of times A occured 

          # of times trial was repeated 

Ex:predicting a face-up/down thumb tack 

Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring.  If event A can occur in s of these n ways, then

P(A)= s = number of ways A can occur

        n   # of different simple events 

Ex: predicting a dice 

P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

Ex: meteorologist's predictions 

As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability.

(The greater the sample number, the better the approximate probability.)  

zero

Ex: Thanksgiving falling on a Wednesday 

One

Ex: Thanksgiving falling on a Thursday 

All outcomes in which the original event does not occur.

Either as exact fractions/decimals or

rounded decimals showing three significant digits.

Ex: 3/5 or 0.0215 

  The ratio P(A)/P(A), usually

expressed in the form of a:b (read as "a to b").

The reciprocal of the actual odds against an event.

Ex: If Odds Against are 1:2, then

Odds in Favor are 2:1. 

Ratio of net profit (if you win) to the amount bet.

payoff odds against event A =

(net profit) : (amount bet) 

any event combining 2 or more simple events

P(A or B) =P (in a single trial, event A occurs or event B occurs or they both occur) 

It is impossible for an event and its complement to occur at the same time.

    P(A)and P(A) are disjoint 

P(A)+P(A) = 1

          P(A)=1-P(A) 

 P(A)=1-P(A) 

Rule for determining the probability that event A will occur on one trial and event B will occur on a second trial

P(A and B) 

The probability of an event, given that some other event as already occured.

When an occurrance from one event does not affect the probability of the occurance of the other.

If A and B are not independent, they are dependent. 

P(A and B) = P(A) • P(B♥A)

"one or more"

The complement of getting at least one item of a particular type is that you get no items of that type. 

calculate the probability of none, then subtract that result from 1.  That is,

P(at least one) = 1 – P(none) 

 A probability obtained with the additional information that some other event has already occurred.  P(B ♥ A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of events A and B both occurring by the probability of event A:

P(B♥A)= P(A and B)/ P(A)

The factorial symbol ! denotes the product of decreasing positive whole numbers.  
For example, 

4! = 4*3*2*1 = 24 

Requirements:
There are n different items available.  (This rule does not apply if some of the items are identical to others.)

We select r of the n items (without replacement).

We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.)