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.
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.
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.
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 .
There are 2 boxes representing a sum of 3 so the
probability of getting a sum of 3 is 2/36.
P(sum of 4 on a pair of dice) = 3/36
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
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
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
Weather Example: You ask someone the chance of rain 20 days from today. They guess 80%. (Some people will have better guesses than others.)
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