10.1 Applying the Counting Principle and Permutations (helps you count up the number of possibilities!)


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1 10.1 Applying the Counting Principle and Permutations (helps you count up the number of possibilities!) Example 1: Pizza You are buying a pizza. You have a choice of 3 crusts, 4 cheeses, 5 meat toppings, and 8 vegetable toppings. How many different pizzas with one crust, one cheese, one meat, and one vegetable are possible? Ask yourself: WHAT DECISIONS WOULD YOU HAVE TO MAKE TO CREATE A PIZZA? (EACH DECISION IS CALLED AN EVENT!) HOW MANY CHOICES YOU WOULD HAVE FOR EACH EVENT? Event 1: Picking a crust Event 2: Picking a cheese Event 3: Picking a meat Event 4: Picking a veggie Total Number of Possibilities: FUNDAMENTAL COUNTING PRINCIPLE Our method for calculating the number of possibilities of simple events. Events: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is. Example 2: License Plates The standard configuration for a Texas license plate is 1 letter, followed by 2 digits, followed by 3 letters. How many different license plates are possible: a. if letters and numbers can be repeated? b. if letters and numbers cannot be repeated? Ask yourself: WHAT WOULD THIS LICENSE PLATE LOOK LIKE? HOW MANY CHOICES DO YOU HAVE FOR EACH EVENT? Number Choices: Letter Choices: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a. Can Repeat b. CanNOT Repeat Total Number of Possibilities: Total Number of Possibilities:
2 Example 3: Playoffs: Five teams are competing in a baseball tournament. How many different ways can the baseball teams finish the competition? Cubs Red Sox Yankees Angels Reds Event 1: 1 st Place Event 2: 2 nd Place Event 3: 3 rd Place Event 4: 4 th Place Event 5: 5 th Place Total Number of Possibilities: How could we do this in the calculator? What is this called? MATH, PRB, 4:! 14! 11! 2! 5! (9 2)! The goal: to find the total number of orders of objects that are possible What to consider: When we have a number of objects n that we would like to arrange or place in r ways or positions, we use a permutation. In a PERMUTATION, we say order matters because a different order creates a new possibility. PERMUTATION n = total number of people/objects r = how many places or positions we are putting them into which can be written n! P n r, which does this for us ( n r)! TI 83/84: 1: Enter your n value  your total number 2: MATH 3: use the right cursor to go over to PRB (probability) 4: Go down to 2: npr 5: Enter your r value 6: Hit ENTER
3 Example 4: Let s say you have 7 friends: 1. Allie 2. Lorenzo 3. Charlie 4. Darica 5. Evan 6. Frank 7. Jamal You decide to find out how many ways you could take all 7 of them and rank them in order of how good of a friend they are (first and second best friend). In other words, how many different orders are possible if we took all 7 of your friends and ranked them 1 st and 2 nd best friend? Example 5: You have chosen 7 classes for your schedule next year. How many different orders are possible? n = r = n = total number of people/objects r = how many places or positions we are putting them into PERMUTATIONS WITH REPETITION The number of distinguishable orders/permutations of n objects where one object is repeated s times, another is repeated s 2 times, and so on, is: s! n! s s 1 2 k! How many ways can the word SEMESTER be arranged? CALIFORNIA S E M T R
4 Let s say you have 7 friends: 10.2 use combinations 1. Allie 2. Lorenzo 3. Charlie 4. Darica 5. Evan 6. Frank 7. Jamal You are all in the same class together. The class has a group project coming up and your teacher tells you to assemble yourselves in groups of 2. You wonder, with all 7 people, how many different minigroups of 2 could be created? does the order matter here? Are you creating orders or groups? We call this a COMBINATION, where we take a group of objects and see how many minigroups of a certain size we can create. COMBINATION n = total number of people/objects r = the size of the group we are putting them into which can be written n C r, which does this for us: n! ( n r )! r! Find the number of possible 5card hands that contain the cards specified. The cards are taken from a standard 52card deck. Example 2: 5 black cards n = total number of ( ) r = the size of the group we are putting them into ( ) C = 5 black cards Example 3: 1 ace and 4 face cards (Jack, Queen, King) 1 ace 4 face cards
5 Example 4: 3 Hearts and 2 nonhearts 3 Hearts 2 nonhearts Example 5: At most 1 diamond Could have 5 card hands with either: 1 diamond: 1 Diamond 4 nondiamonds 0 diamond: 0 Diamond 5 nondiamonds Example 6: You are picking 7 books from a stack of 32. If the order of the books you choose is not important, how many different groups of 7 books are possible? Example 7: The new releases consist of 12 comedies, 7 drama, 5 suspense, and 9 family movies. You want exactly 2 comedies and 3 family movies. How many possibilities are there for renting 2 comedies and 3 family movies? Event 1: Picking a comedy: Event 2: Picking a family movie: Total # of possibilities:
6 10.3 Define and use Probability and odds To find the probability that an event will occur, we use: PROBABILITY Success Total Our answer will be a number from 0 to 1 that indicates the likelihood that the event will occur. We may write this as a fraction, a decimal, or a percent. Example 1: The integers 1 through 20 are written on separate pieces of paper and placed in a hat. Find the probability of: a. choosing a perfect square b. choosing a factor of 40 Example 2: A single card is drawn from a standard deck of 52 cards. Find: (a) the probability of drawing a black King Poss. of drawing a Black King Poss. of drawing any card (b) the probability of drawing a card other than a Spade
7 Example 3: A candy jar contains 16 chocolate eggs, 2 Reese s eggs, and 7 Coconut Hershey Kisses. If you reach your hand into the jar once and pull out one piece of candy, find: P (selecting 1non  egg candy) Example 4: Same candy jar: Your little brother walks up to the candy jar, reaches his hand down in one time, and decides to grab two pieces of candy. Find the probability that he selects 2 Coconut Hershey Kisses. P ( selecting 2 CHK) Poss.of drawing 2 CHK Poss.of drawing any 2 pieces of candy How many chances are there of drawing 2 CHK?? (Be careful here. Think about it out of a group of 7 kisses he is selecting groups of 2.) How many different possibilities could come out of that? (Be careful here. Think about it out of a group of 25 candies he is selecting groups of 2.) To find the odds that an event will occur, we use: ODDS Success Failures Odds can be written as a fraction, a ratio 3:5, or 3 to 5. Odds that event A will occur: : 5 3 to 5 Example 5: You randomly choose a drink from a cooler. The cooler contains 26 pop cans. There are: 13 Sunkist, 5 Pepsi, 3 Mountain Dew, and 5 Sprite. (a) Find the odds in favor of drawing a Mountain Dew or Sprite can. (b) Find the odds against drawing a Sunkist can.
8 EXPERIMENTAL PROBABILITY When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is: Number of trials where A occurs Total number of trials Example 6: Exam Grades Exam grades of students in a history class are shown in the bar graph. Find the probability that a randomly chosen student in this history class received a C or better.
9 OR 10.4 Find Probabilities of Disjoint & Overlapping Events A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a 10 or a face card? DISJOINT EVENTS (ALSO CALLED MUTUALLY EXCLUSIVE) + = Probability of a 10 Probability of a face card Total Prob. In general, two events are disjoint, meaning they are separate events, when their outcomes have nothing in common, and they are connected by the word OR. To find the probability: P(A or B) = _P(A) + P(B)_ EXAMPLE 1: You roll a sixsided number cube. What is the probability of rolling 1 or an even number? + = Event A: roll a 1 Event B: roll even??how do we know that these events are disjoint?? Events A and B are disjoint. Find P(A or B). P(A) = 0.85, P(B) = 0.05 Events A and B are disjoint. Find P(A or B). P(A) = 2 1, P(B) = 5 1 OVERLAPPING You roll a sixsided number cube. What is the probability of rolling an odd number or a number less than 3? + = roll an odd number roll a number less than 3 Overlap Total Prob. odd AND less than 3 P(odd or less than 3) In general, two events are overlapping when they are separate events that have some outcomes in common, and are joined by the word OR. To find the probability: EVENTS P(A or B) = _P(A) + P(B) P(A and B)_
10 EXAMPLE 2: A card is randomly selected from a standard deck of 52 cards. Find the probability that it is red OR a face card. FIND THE INDICATED PROBABILITY P(A) =, P(B) = P(A or B) = 11 + = A B Overlap Total Prob. P(A and B) P(A or B) P(A and B) = 4. P(A) = 0.5 P(B) = 0.35 P(A and B) = 0.2 P(A or B) = + = A B Overlap Total Prob. P(A and B) P(A or B) IN NUMBERS 3 AND 4, ARE EVENTS A AND B DISJOINT? EXPLAIN WHY OR WHY NOT? Determine whether the following events would be Disjoint or Overlapping. DISJOINT OVERLAPPING? DISJOINT OVERLAPPING? P(selecting a Queen or a card under 10) P(selecting black card or an even numbered card) DISJOINT OVERLAPPING? P(selecting a person with contacts or a person on a sports team) DISJOINT OVERLAPPING? P(selecting a person wearing jeans or a person who s 16) COMPLEMENT The event A, called the complement of event A, consists of all outcomes that are not in A. The notation A is read as A bar. The probability of the complement of A is P( A ) = 1 P(A). Tomorrow there is a 63% probability of rain So P(not rain) = 1
11 AND 10.5 Find Probabilities of independent and dependent events In general, events are independent if the occurrence of one has no effect on the occurrence of the other, and if the purpose is to find the probability that both the first AND the second event will occur. P(A and B) = _P(A) P(B)_ INDEPENDENT EVENTS Every morning, one student in a class of 24 students is randomly chosen to take attendance. What is the probability that the same student will be chosen three days in a row? Are these independent events? = P(Day 1) P(Day 2) P(Day 3) Total Prob. EXAMPLE 1: For a fundraiser, a class sells 150 tickets for a mall gift certificate and 200 raffle tickets for a booklet of movie passes. You buy 5 raffle tickets for each price. What is the probability that you win both prizes? Why are these independent events? P(win certificate AND win booklet) = P(cert.) P(booklet) Total Prob. Events A and B are independent. P(A) = 0.5 P(B) = 0.25 P(A and B) = Events A and B are independent. P(A) = P(B) = 0.6 P(A and B) = 0.15
12 Two events A and B are dependent if the occurrence of one affects the occurrence of the other, and if the purpose is to find the probability that both the first AND the second event will occur. DEPENDENT EVENTS The probability that B will occur given that A has already occurred is called the conditional probability of B. In other words, B s probability depends on what happened with A. We use P(B A) to denote B s probability. P(A and B) = _P(A) P(B A) EXAMPLE 2: The host of a game show is drawing chips from a bag one at a time to determine the prizes for which contestants will play. Of the 10 chips in the bag, 6 are labeled TV, 3 are labeled V for a vacation to Italy, and 1 is labeled GT for the Mustang GT you could win. If the host draws the chips at random and does not replace them, find each probability. a. P(a TV, then a car) TV TV TV TV TV TV V V V G T b. P(a television, then another television) EXAMPLE 3: Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a diamond, a club, and another diamond in that order. EXAMPLE 4: Three cards are drawn from a standard deck of cards with replacement. Find the probability of drawing a diamond, a club, and another diamond in that order. P(A) = 0.8 P(B A ) = P(A and B) = 0.32 P(A) = 0.3 P(B A ) = 0.6 P(A and B) =
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