Probability Warm-Up 1 (Skills Review) Directions Solve to the best of your ability. (1) Graph the line y = 3x 2. (2) 4 3 = (3) 4 9 + 6 7 = (4) Solve for x: 4 5 x 8 = 12? (5) Solve for x: 4(x 6) 3 = 12? (6) Literal equations: If d = rt, solve for r.
Permutations, Combinations, and the Counting Principle Essential Question: How can large groups of things best be organized and counted? Academic Vocabulary: Permutations, combinations, factorial, arrangements, sample space Questions: What is the fundamental counting principal? One of the first things you need to understand in order to succeed with probability is how to properly count all the possible outcomes for a given situation. Outcomes are counted with variations on factorial multiplication:! 1 2 1. Example: How many different ways can you order questions on a five question quiz? Example: How many different ways can you order questions on an eight question quiz? Factorial values get very large very fast. That s why we use an exclamation point! The counting principal can be extended to count multiple groups of things. Example: How many different outfits can you make from three different shirts, five pairs of pants, and four hats? Example: How many different outfits can you make from six different shirts, four pairs of pants, and two hats? What is a permutation? Sometimes we want to know how many ways we can pick a limited number of items from a larger group. Example: How many ways can you create a four class period schedule from a pool of twelve possible classes? Example: How many different five number passwords can you create from the numbers zero through nine is repetitions are not allowed? Summary:
Permutations, Combinations, and the Counting Principle Questions: What is a combination? The previous two examples are called permutation problems. Officially, the permutation formula looks like this:!. A further extension of the counting! principle contends with the fact that order does not always matter. That is, under some circumstances we might need to consider 12345 as functionally equivalent to 54321. Example: How many different ways can four students be chosen from a class of thirty? Example: Three out of twenty students will be selected to represent Cedar Ridge. How many different ways can the three be chosen? These problems are called combinations problems or problems of choice. They differ from permutations because the order in which things are chosen does not influence the total number of possibilities. Officially, the combination formula looks like this:!.!! Discerning which method to apply or how to properly adjust the correct method is what makes probability seem difficult. As with most things in life, the devil is in the details. Summary:
Permutations, Combinations, and the Counting Principle Variations on Counting Principal Problems Sometimes the number of possibilities is small enough to write them all out. 1) A spinner can land on red, blue, yellow or green. You flip a coin and then spin the spinner. List the possible outcomes. But usually a counting technique must be applied. 2) How many different ways can you arrange the letters in the word WASHTUB? 3) How many different ways can you arrange the letters in the word MISSISSIPPI? How did you deal with the repeated letters in Mississippi? Allowing for repetition also changes our approach slightly. 4) A simple password is five characters long, chosen from a group of eight possible characters, repeats are allowed. How many possible passwords can you make? What mathematical operation does question four involve? Sometimes we must apply a technique more than once. 5) A six person student council will be chosen from a group of ten seniors and eight juniors. The council must be composed of four seniors and two juniors. How many possible student councils can be created? What do all counting principal problems have in common? Summary:
Permutations, Combinations, and The Counting Principle Practice Problems e K2C0E1n8G xkkuktjag TSToOfGtiwSaxr\ey tlglvch.h W razlqlx urdihgmhdtfsy _rrekswewrbvdemd\. Represent the sample space by writing out each possibility. 1) When a button is pressed, a computer program outputs a random odd number greater than 1 and less than 9. You press the button twice. 2) A sandwich shop has four types of sandwiches: ham, turkey, chicken, and PB&J. Each sandwich can be ordered with white bread or multi-grain bread. Find the number of possible outcomes in the sample space. 3) A sandwich shop has three types of sandwiches: ham, turkey, and chicken. Each sandwich can be ordered with white bread or multi-grain bread. Customers can add any combination of the four available toppings 4) A spinner can land on either red, blue, or green. You spin six times. 5) A basketball player attempts seven free throws. Each attempt results in a score or a miss. 6) A spinner can land on either red or blue. You spin seven times. 7) You flip a coin twelve times. 8) Seven books need to be placed on a shelf. You randomly arrange the books on the shelf from left to right. Find the number of unique permutations of the letters in each word. 9) CALCULUS 10) VANQUISH 11) NUMBER 12) PARALLEL V q2w0]1^8t lkvubtkao ssxocfhtywbapreea altldc\.x g GAOlblg trxihgmhttqsc crpemsee[rpvre^db.b T xmvaxdjed gwniktyha YIunJfciwnTiOtXew BAElxgYelblrxag o2^. -1- Worksheet by Kuta Software LLC
State if each scenario involves a permutation or a combination. Then find the number of possibilities. 13) Heather and Bill are planning trips to three countries this year. There are 7 countries they would like to visit. They are deciding which countries to skip. 14) The student body of 50 students wants to elect a president, vice president, secretary, and treasurer. 15) A group of 12 people need to take an elevator to the top floor. They will go in groups of six. They are deciding who will take the elevator on its second trip. 16) 5 out of 20 students will ride in a car instead of a van 17) Selecting which seven players will be in the batting order on a 8 person team. 18) There are 50 applicants for two Systems Engineer positions. 19) Selecting which seven players will be in the batting order on a 10 person team. 20) You are setting the combination on a three-digit lock. You want to use the numbers 123 but don't care what order they are in. 21) A company has four available jobs for thirty applicants. Twelve of the applicants are women while the other eighteen applicants are men. The company wants to hire two women and two men. How many different hiring combinations are possible? l d2f0j1n8e JKuuitYaG MSso[fhtbwqaHrte_ ULULyCY.f Q `AElnlZ [rqijgkhetusn CryeYsSeHrTvHecdG.a L emmaddper \wkiethhx [IwncfFi`nbiGtZen ZAflkgNeKbRrCaC R2j. -2- Worksheet by Kuta Software LLC