Unit 7: ProbabilityLesson 2Tree Diagrams, fundamental counting principle, sample space lists and tables, permutation, combination, factorials

The Standard:

MCC9-12.S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Learning Target

I can identify and represent sample spaces

Vocabulary:

Tree diagrams Fundamental counting principle Permutation Combination factorial

Sample Spaces: There are several ways to determine

the number of possible outcomes for an event:

The fundamental counting principle: multiply the choices

Factorials: when all the items are to be arranged

Permutations: when order matters Combinations: when you are creating

groups and order is not important

Factorials: Factorials are very simple things.

They're just products, indicated by an exclamation mark. For instance, "four factorial" is written as "4!" and means 1×2×3×4 = 24. In general, n! ("enn factorial") means the product of all the whole numbers from 1 to n; that is, n! = 1×2×3×...×n.

(For various reasons, 0! is defined to be equal to 1, not 0. Memorize this now: 0! = 1.)

Fundamental Counting Principle

If you have 3 tops, 4 pairs of slacks, and 2 belts, how many different outfits can you make?

Permutation

Permutations: when order matters PERMUTATIONS

A permutation is "a re-arrangement of elements of a set".

So, what does this mean? It means a permutation is ONLY interested in re-arranging the elements of the set... Any duplications of the collected elements in different orders is fine.

A permutation therefore tends to be a large number.

Permutations example: Example:

Taking the 4 letters, ABCD, write down all the permutations of 3 of these letters:

ABC BAC CAB DABACB BCA CBA DBAABD BAD CAD DACADB BDA CDA DCAACD BCD CBD DBCADC BDC CDB DCB

--> there are 24 permutations. In other words, just taking each letter and collecting them into sets of 3 from the 4 and writing them out, gives 24 variations. Done.

Here, if you like, the order matters, since ABC is different to ACB and different to BCA and different to CAB etc. Permutations see these as all different answers

Combinations: COMBINATIONS

A combination is "one or more elements selected from a set without regard to the order"

The "without regard" means that the collection matters rather than order in combinations, so in the above example, the fact we ABC, ACB, BAC, BCA, CAB, CBA... for combinations, these are all 1 combination of letters A, B and C.

So, questions concerning picking a team of 5 people from a squad of 11... you would need combinations, since it is having "Bert, Ernie, Fred, Bill and Bob" that matters, not the fact that you have so many different permutations of these 5 people.

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Combinations Example Example:

Taking the 4 letters, ABCD, write down all the combinations of 3 of these letters:

ABC ABD ACD BCD

--> there are just 4 combinations. You cannot pick any other 3 letters from ABCD, that is not part of the above 4 combinations. It is enlightening to see the letter missing in each: in order we have "no D", "no C", "no B" and finally "no A"... this sometimes helps you to "see" all the possible answers.

To calculate combinations, it is a 2-stage process: 1. you calculate all the equivalent permutations first.2. you then correct this list by cutting out any duplicates.

As you can see from this, combinations are a subset of Permutations