Calculating Probability, Combinations and Permutations

Fundamental Counting Principle A.K.A. Product Rule: Multiply events together to get the total number of outcomes. Ex A: 1 Coin: 2 outcomes 2 Coins: 2 x 2 =4 outcomes 3 Coins: 2 x 2 x 2 =8 Used to figure out the number of outcomes in a probability problem. Your Turn: The Sandwich shop offers a $5 combo. Choose 1 from each category: There are 3 sandwich choices, 2 sides, and 3 beverages. 3 x 2 x 3 = 18 http://www.statisticshowto.com/fundamental-counting-principle/

Permutations Are “PICKY”. Order Matters. It is the number ways to organize outcomes when ORDER MATTERS Think about an organized List *remember the coin flip problem? You found 8 permutations for the sample space! Examples: A Combination Lock really should be called a PERMUTATION LOCK i.e. Your combo is 12-30-18, so 30-18- 12 does not work!

How many options? Calculating Permutations Suppose we have 5 people in a race. How many ways can Gold, Silver and Bronze be awarded? Options? Make an organized list A,B,C,D,E __, __, __ But Wait, these are just from the first 3 letters! What happens next? ABC ACB BAC BCA CAB CBA https://betterexplained.com/articles/easy-permutations-and-combinations/

How many options? Calculating Permutations Suppose we have 5 people in a race. How many ways can Gold, Silver and Bronze be awarded? Options? Make an organized list A,B,C,D,E __, __, __ How about BCD, BDC, CBD, CDB, DBC, DCB Hmm. Let’s try that counting principle… 5 x 4 x 3 = 60 So, we have: ABC ACB BAC BCA CAB CBA

Is there a Formula for this stuff? N! Factorial. The product of an integer and all integers below it

Combinations Are “Easy going” Order Does not Matter But, how many ways can I PERMUTE those 3 ingredients? Are “Easy going” Order Does not Matter It is the number of outcomes when order doesn’t matter 3 x 2 x 1 = 3! (Three Factorial), or 6 ways. BUT it’s really ONE smoothie… So, let’s get back to that... : Example A strawberry, apple, banana smoothie tastes the same as a banana, strawberry, apple one! 3 Choose 3 = 1

Try it! So, how many ways can I PERMUTE those 3 ingredients? Make an organized list or tree diagram SAB, SBA ABS, ASB BAS, BSA Use the Formula 3! (3 - 3)! = 6 9/14

Calculating Combinations Consider the First problem. What if we DIDN’T care about the order? Then these results would be the SAME! ABC ACB BAC BCA CAB CBA The question is NOW: How can I choose ANY 3 people out of the 5? Now how many ways can 3 people can arranged without order? Can we list them? https://www.khanacademy.org/math/precalculus/prob-comb/combinations/v/introduction-to-combinations https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php

Calculating Combinations Consider the First problem. What if we DIDN’T care about the order? Then these results would be the SAME! ABC ACB BAC BCA CAB CBA The question is NOW: How can I choose 3 people out of the 5? Now how many ways can 3 people can arranged without order? Can we list them? https://www.khanacademy.org/math/precalculus/prob-comb/combinations/v/introduction-to-combinations https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php

Combination Formula

9/15 Do Now: In Notebooks Use a tree diagram to determine how many ways you can choose an outfit based on 1 pair of Black shoes, Blue pants vs. black pants, and a blue or white shirt. Is this a Permutation or Combination or Neither? 3. If possible, Write the formula to solve this problem, then use a calculator to check!

CLASSWORK Text pg. 969 #16-19 Use the following protocols: Work independently first 2-3 min Silently, using your resources Next you may consult with an elbow partner (in YOUR ROW) Talk on task Talk only with partner(s) Keep voices at level 1 Minutes 7-10 finish the assignment independently