PowerPoint Presentation

Counting Principles and Probability

We equate probability with what are my chances.

Counting Principles and ProbabilityWe must learn to count beforewe can calculate probability!Counting PrinciplesHow many outcomes are possible?How many unique outcomes are possible?How many outcomes that make sense are possible?

Your familiar with the situation: You have 4 tops, 5 hats and 6 pairs of pants. Assuming everything matches how many different outfits can you make? This is a CHAIN of events each with its own number of possible outcomes.

So the total number of possible outcomes would be 456 which is equal to 120 outcomes.

FUNDAMENTAL COUNTING PRINCIPLE If you have a chain of evens that each have m outcomes the possible outcomes of the entire chain would be the product of the individual events that make up this chain.Counting Principles

How many possible ways are there to get a sum of 7?How many possible ways are there to get a sum of 7?

Counting PrinciplesHow many ways can we draw a card from the bag containing numbers 1 -14 REPLACE it, then draw again AND get those two numbers to sum to 17?12345678910111213141431341251161079812 waysHow many ways can we draw a card from the bag containing numbers 1 -14 WITHOUT REPLACEMENT, then draw again AND get those two numbers to sum to 17?1234567891011121314143134125116107986 waysFOR EXAMPLE: How many 3 element lottery tickets can be created if the first element must be a number from 1- 5, the second must be a letter from A-G and the third must be a number from 10-12? 12345GFEDCBA1211105A3105 ways573FOR EXAMPLE: How many 3 element lottery tickets can be created if the numbers must be picked are 1- 14 in any order, no repeats?12345678910111213141413122184 waysFOR EXAMPLE: How many 3 element lottery tickets can be created if the numbers must be picked are 1- 14 in any order repeats okay?12345678910111213141414142744 waysWhat if you are counting outcomes and order matters? Like in a race (first, second, third?)When ORDER MATTERS it is called PERMUTATION.

How many different finishes can there be for the Sausages?

543How many different finishes can there be for the Sausages? This could also be determined using the formula

nPrHow many different finishes can there be for the Sausages? This could also be determined using the formula

5P3How many different finishes can there be for the Sausages? This could also be determined using the formula

5P360BANANA

How many different combinations of the word BANANA can you make?BANANA

6543217206!But there would be repeats in there because you couldnt tell one A from another A. So if we want DISTINGUISHABLE outcomes (DISTINGUISHABLE PERMUTATION) wed have to get rid of repeats.BANANA

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Remember: When order matters (when its more important to be first than second) its permutation.

If order doesnt matter its a combination!Picking volunteers from a larger groupPicking committee members from a larger groupPermutation or Combination?Picking prom courtCSitting in open seating at a concertCLining up for school lunchPChoosing the president and vp of junior classPPoker hand.C12 person swim team from a team of 30C6 person relay teamPIf order doesnt matter its a combination!nCrA standard poker hand consists of 5 cards dealt from a deck of 52. How many different poker hands are possible.52C52,598,960You are forming a 12 member team from 10 girls and 15 boys. The team must have five girls and seven boys. How many different 12 member teams are possible?10C5 15C71,621,620