Classical Definition of Probability

Given n equally likely outcomes, where s  represents the number of successful outcomes, and f represents the number of failing outcomes,  s + f = n. The probability of success is s/n . The probability of failure is f/n . The probability of success  plus the probability of failure is 1 or 100%         s/n  +  f/n  = (s + f)/n  =  (s + f)/(s +f)  =  1  .

[ We will use a capital P, for Probability, followed by a description of success in parenthesis and then equals s/n ]

Box of Marbles Example:   A box contains 3 red marbles, 1 blue marble, and 4 yellow marbles. One marble is drawn at random. There are now 8 equally likely marbles that can be drawn.
P(draw one of the eight marbles and it is red)  =  3/8
P(draw one of the eight marbles and it is blue)  =  1/8
P(draw one of the eight marbles and it is yellow)  =  4/8
P(draw one of the eight marbles and it is not red)  =  5/8
P(draw one of the eight marbles and it is red or blue)  =  4/8

Fair Die Tossed Fairly Example: A fair die used in numerous games has 6 sides that should have the same area on each side and are marked 1, 2, 3, 4, 5, and 6 respectively with dots or numbers. Each side should be equally likely to be shown on top when tossed.
P(a one turns up)  =  1/6
P(a two turns up)  =  1/6
P(a three turns up)  =  1/6
P(a four turns up)  =  1/6
P(a five turns up)  =  1/6
P(a six turns up)  =  1/6
P(a five or six turns up)  =  2/6

Fair Pair of Dice Tossed Fairly Example: In many games a pair of dice is tossed. The two numbers that turn up are added for a sum of the pair of dice. The Classical Definition can be applied if tossing a pair is considered as tossing a first die and then a second die,  or, tossing one die first and then tossing the same die again for the second toss.  Or, if using the Dice-in-Dice, we consider the first toss or die as the large one and the second toss the small white one. An equally likely outcome table is shown below for a die tossed twice or the Dice-in-Dice tossed one is large and one is small.

Small
(or second die)
Number on top
 1 2 3 4 5 6
Large
(or first die)
Number on top
 1 2 3 4 5 6
 1+1 1+2 1+3 1+4 1+5 1+6 2+1 2+2 2+3 2+4 2+5 2+6 3+1 3+2 3+3 3+4 3+5 3+6 4+1 4+2 4+3 4+4 4+5 4+6 5+1 5+2 5+3 5+4 5+5 5+6 6+1 6+2 6+3 6+4 6+5 6+6

There are 36 equally likely outcomes where the sum in each of the 36 boxes represents the outcome for that box. To get a sum of 2 for a fair pair of dice tossed fairly, there is only one box (upper left corner) which has a sum of 2. Since there is only one outcome out of 36 that are equally likely, then the probability of getting a sum of 2 is 1/36 .
P(sum of 2 on a pair of dice)  =  1/36

There are 2 boxes representing a sum of 3 so the probability of getting a sum of 3 is 2/36.
P(sum of 3 on a pair of dice)  =  2/36

P(sum of 4 on a pair of dice)  =  3/36
P(sum of 5 on a pair of dice)  =  4/36
P(sum of 6 on a pair of dice)  =  5/36
P(sum of 7 on a pair of dice)  =  6/36

P(sum of 8 on a pair of dice)  =  5/36
P(sum of 9 on a pair of dice)  =  4/36
P(sum of 10 on a pair of dice)  =  3/36
P(sum of 11 on a pair of dice)  =  2/36
P(sum of 12 on a pair of dice)  =  1/36
P(doubles on a pair of dice)  =  6/36
P(less than a sum of 5 on a pair of dice)  =  6/36

Note:  Since the probability is defined in terms of equally likely outcomes, this Classical Definition is circular, nevertheless practical.

 A knucklebone has rounded ends and when tossed will land on one of 4 sides (2 narrow sides and 2 wide ones). This is a 4-sided die used for thousands of years. And sides that turn up are not equally-likely, so the Classical Definition does not work.    Theoretical Definition of Probability  A trial is conducted which can be repeated, and each trial results in either success or failure, but not both. As the trials are repeated, the ratio of the number of successes, s, to the number of trials, n, are observed. As the number of trials are conducted forever,  s/n  will get closer and closer to the actual value of the probability of success. This definition may be useful in theoretical math, but as you can see it is not applied in a practical way since we don't perceive repeating trials forever. Knucklebone Example:   Paint one of the four sides of a knucklebone red. Toss it 10 times, or trials, recording the number of successes. Say, 3 out of 10 tosses turn up red. Toss it 90 more trials, say 18 out of all 100 times red turns up. Assume you continue to toss it and record the successes. ...3/10...18/100...173/1000...s/n... ---->    Probability Value If the knucklebone does not wear out and we don't, then  s/n  , relative frequency of success out of n trials, will approach the actual Probability Value. We don't know what this value is since this would take forever since n approaches infinity.
 Relative Frequency Definition of Probability  A trial is conducted which can be repeated, and each trial results in either success or failure, but not both. As the trials are repeated, the ratio of the number of successes, s, to the number of trials, n, are conducted for some large value of n. When you stop at some value of n you take that relative frequency,  s/n,  as the probability value (approximately). Knucklebone Example:   Paint one of the four sides of a knucklebone red. Toss it 10 times, or trials, recording the number of successes. Say, 3 out of 10 tosses turn up red. Toss it 90 more trials, say 18 out of all 100 times red turns up. You stop tossing it and decide to use  18/100  as the Probability of getting red. Opinion Poll Example:   It is desired to know what the probability is that someone in a particular population favors blue over other colors. We pick at random 150 people in this population, and 39 of the 150 favor blue. The probability of someone in this population favoring blue is the relative frequency 39/150  =  0.26 . (We generally say about 26% of this population favor blue.)
 All of the definitions of probability result in values between 0 and 1. Rare outcomes have probabilities close to 0, and very common outcomes have probabilities close to 1. Sometimes this is expressed in percentages, 0% to 100%. Subjective Definition of Probability  A simple subjective definition of the probability of a particular outcome is a guess. (Sometimes called an educated guess or a degree of belief or judgment. A probability value is unconsciously or consciously arrived at and even may be biased. Statistical Trends alone may predict a particular outcome but sometimes a  person has more information and can make a better prediction.)   Weather Example:   You ask someone the chance of rain 20 days from today. They guess 80%. (Some people will have better guesses than others.)