# Probability

Base Definitions
Experiment is a controlled operation that yields a set of results
Outcome is each individual result
Event is a collection of the results
Equally likely outcomes is when each outcome has the same chance of happening
For example, if I were to roll a fair 6-sided die, this would be my experiment. I could roll it 20 times, and record what it lands on each time, each of these would be my outcomes. My Event would be all the times it landed on 3.
Probability is the study of the chance a situation will occur. There are several types of probability, including:
Theoretical Probability is when the probability is calculated based on what is thought will happen based on equally likely outcomes, sometimes called Mathematical Probability.
Theoretical Probability is the Number of Outcomes favorable to the Event divided by the Total possible Outcomes., ie P(E) = N/T.
Using the die above, the Theoretical probability would be P(3) = 1/6, since there is only 1 out of the 6 sides with a 3 on it.
Empirical Probability is when the probability is calculated based on what has happened before, sometimes called Posterior Probability or Relative Frequency, although this last one is technically a little different.
Empirical Probability is the Number of time the Event Occurred divided by the Total number of times the experiment was performed, ie P(E) = N/T.
Again, using the die above suppose when I rolled the die it landed on 3, 7 times, the Empirical probability would be P(3) = 7/20, reminder, we often will be reducing the fraction or converting it to a percent.
Subjective Probability often refers to a personal opinion of an event happening.
For example, I have cats and I also knit with yarn. There is a high probability that when I sit down to knit, at least one of the cats with try to help me with the yarn.
Probability will always be between 0 & 1, inclusive, i.e. 0 ≤ P(E) ≤ 1
So the probability something won't happen is 0, i.e. P(rolling 9 on a standard 6-sided die) = 0
So the probability something will happen is 1, i.e. P(the sun will rise tomorrow) = 1
The sum of all probabilities is 1
The probability of something not happening is P(not E) = 1 – P(E)
The probability an event happens at least once is P(E happens at least once) = 1 – P(E never happens)
Law of Large Numbers states that the more times an experiment is performed the closer the Empirical Probability gets to the Theoretical Probability. This does not mean that 'your number will eventually come up'.
The following video goes over most of the above definitions, and a couple of others, and is about 4.5 minutes.
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The following video goes over Theoretical Probability, Empirical Probability, and Odds use of the Formulae and is about 4 minutes.
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Odds Against is the ratio of the probability of failure to the probability of success, and is written with a colon for the answer.
Odds Against = P(Failure)/P(Success), so re-examining the "rolling a 3" from above. The P(success) = 7/20, the P(failure) = 13/20, so the Odds Against are 13:7. It may be necessary to review division of fractions.
The following video goes over Odds Against and is about 3 minutes.
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Odds In Favor is the ratio of the probability of success to the probability of failure, and is written with a colon for the answer.
Odds in Favor = P(Success)/P(Failure), so re-examining the "rolling a 3" from above. The P(success) = 7/20, the P(failure) = 13/20, so the Odds in Favor are 7:13.
The following video goes over Odds In Favor and is about 2 minutes.
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Expected Value is the sum of the products of the probabilities with their values.
This time the formula may be easier to read, but we will keep it simple. We could have any number of probabilites, but we will just use 6. Where Pn is the nth probability, and An is the nth amount or value.
Expected Value = P1*A1+P2*A2+P3*A3+P4*A4+P5*A5+P6*A6
The following video goes over Expected Value and is about 4.5 minutes.
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Fair Price is the price that really should be charged
Fair Price = Expected Value + Amount Paid
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More Definitions and Topics
Sample Space is the set of all possible outcomes in an experiment
Sample Point is any one of the outcomes in the Sample Space
Tree Diagram is one way of listing the Sample Space
The following video goes over Tree Diagrams and is about 5 minutes.
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Compound Probability is when the probability of more than one outcome is to be determined
Mutually Exclusive Events are events that can't both happen
Independent Events happen when one event does not affect the other event in any way
Dependent Events happen when one event does affect the other event in some way
With Replacement is if an item is selected and then put back, it may be chosen again
Without Replacement is if an item is selected, but it is not put back
The following video goes over the above definitions and is about 2.75 minutes.
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Counting Principle tells us the number of ways 2, or more, experiments can be performed in order
If there are only two situations, with the first one having M results and the second one having N, then the number of possible results will be M*N
The following video goes over the Counting Principle and is about 2.5 minutes.
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Or Probability is the probability that either event will occur.
P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) in this case is the probability that they both occur at the same time, this will be 0 if they are mutually exclusive.
And Probability is the probability that both events will occur.
P(A and B) = P(A)*P(B)
This time, they are two events, and the concept is similar to the counting principle.
The Or & And Probabilities are similar to the Union and Intersection of Sets.
Permutations examine the number of ways a group of objects can be ordered, if you have n items, you will have n! ways in which they can be ordered.
If you have 10 items, and you want to determine how many different ways they can be ordered, you use this principle again. This time, you will have 10 options for the first spot, 9 for the second, and so on, so you will have 10*9*8*7*6*5*4*3*2*1 or we can use the notiation of !, so we have 10!, which is read as "10 factorial".
However, if you only want to select r items in order out of the n items, you will use this permutation formula: nPr = n!/(n-r)!
The following video goes over this Permutation Formula and is about 4 minutes.
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Combinations are permutations when the order doesn't matter.
That is when you just need to choose a certain number of items from a list. For example, when choosing a combination from a menu, you choose a couple from several, usually 2 from the 7 listed or similar. It doesn't matter which way you list them to the server, they will often be put on the same plate.
When you are choosing r items out of the n items, you will use the combination formula: nCr = n!/[(n-r)!r!]
The following video goes over Combinations and is about 3.5 minutes.
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Definitions and Formula are from numerous years of teaching the topics, but have recently been double checked with:
Borowski, E. J., & Borwein, J. M. (2006). Collins Web-linked dictionary of mathematics. New York, NY: HarperCollins Pub.