#### Transcript Probability

Probability Foldable #1 Do a hotdog bun fold and make 10 cuts Combination Conditional probability Dependent events Events Experimental probability Independent events Mutually exclusive events Permutation Probability Sample space Theoretical probability Vocabulary Words • Combination - a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter. • Conditional probability – the probability of event B given that event A has occurred. • Dependent events – the occurrence of one event affects the probability of the other. Vocabulary Words: These will be on the test • Event – an outcome or set of outcomes. • Experimental probability - the likelihood that the event occurs based on the actual results of an experiment experimental probability= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 • Independent Events – when the occurrence of one event does not affect the probability of the other • Mutually Exclusive Events – events that cannot both occur in the same trial of an experiment. Vocabulary Words • Permutation - a selection of a group of objects in which order is important. • Probability – the measure of how likely an event is to occur. • Sample space – the set of all possible outcomes. • Theoretical probability – the likelihood of an event based on mathematical reasoning 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 P(event)= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Experimental and Theoretical Probability • Outcome – each possible result of a probability experiment or situation. • Event – an outcome or set of outcomes. • Sample space – the set of all possible outcomes • Probability – the measure of how likely an event is to occur • Theoretical probability – the ratio of the number of favorable outcomes to the total number of outcomes. 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 P(event)= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Experimental Probability Experimental probability of an event – the number of times that the event occurs. experimental 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠 probability= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑑𝑜𝑛𝑒 Finding Experimental Probability The table shows the results of a spinner experiment. Find each experimental probability. Number Occurrences 1 6 2 11 3 19 4 14 A. Spinning a 4 14 50 = 7 =28% 25 There were 50 occurrences B. Spinning a number greater than 2 33 = 66% 50 Theoretical Probability • Theoretical probability – the ratio of the number of favorable outcomes to the total number of outcomes. 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 P(event)= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Finding Theoretical Probability A CD has 5 upbeat dance songs and 7 slow ballads. What is the probability that a randomly selected song is an upbeat dance song? P(upbeat dance 5 song)= 12 ≈ 41.7% Probability Distributions and Frequency Tables • Frequency table – a data display that show how often an item appears in a category. • Relative frequency – the ratio of the frequency of the category to the total frequency. relative frequency 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑡𝑒𝑔𝑜𝑟𝑦 = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 • Probability distribution – shows the probability of each possible outcome. Finding Relative Frequencies The results of a survey of students’ music preferences are organized in this frequency table. What is the relative frequency for each type of music? a. classical 1/8 b. Hip hop 7/40 c. Country 1/5 Type of Music Preferred Frequency Rock 10 Hip Hop 7 Country 8 Classical 5 Alternative 6 Other 4 Your Turn The results of a survey of students’ music preferences are organized in this frequency table. What is the relative frequency of preference for rock music? 10 = 10+7+8+5+6+4 = 10 40 = 1 4 Type of Music Preferred Frequency Rock 10 Hip Hop 7 Country 8 Classical 5 Alternative 6 Other 4 Calculating Probability by Using Relative Frequencies A student conducts a probability experiment by tossing 3 coins one after the other. Using the results below, what is the probability that exactly two heads will occur in the next three tosses? Coin Toss Result HHH HHT HTT HTH THH THT TTT TTH Frequency 5 7 9 6 2 9 10 2 Calculating Probability by Using Relative Frequencies Step1: Find the number of times a trial results in exactly two heads. The possible results that show exactly two heads are HHT, HTH, and THH. The frequency of these results is 7+6+2=15 Step 2: Find the total of all the frequencies. 5+7+9+6+2+9+10+2=50 Step 3: Find the relative frequency of ta trial with exactly two heads. relative frequency= 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 𝑡𝑤𝑜 ℎ𝑒𝑎𝑑𝑠 𝑡𝑜𝑡𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 = 15 50 = 3 10 Your Turn A student conducts a probability experiment by spinning the spinner shown. Using the results in the frequency table, what is the probability of the spinner pointing at 4 on the next spin? Spinner Result 1 2 3 4 Frequency 29 32 21 18 18/100=9/50 Finding a Probability Distribution In a recent competition, 50 archers shot 6 arrows each at a target. Three archers hit no bull’s eyes; 5 hit one bull’s eye; 7 hit two bull’s eyes; 7 hit three bull’s eye; 11 hit four bull’s eye; 10 hit five bull’s eye; and 7 hit six bull’s eye. What is the probability distribution for the number of bull’s eyes each archer hit? Finding a Probability Distribution Plan Know Need The possible outcome and the frequency of each outcome Make a frequency table and use relative frequencies to complete the probability distribution. The probabilities of each outcome First, create a frequency table showing all the possible outcomes: 0, 1, 2, 3, 4, 5, or 6 bull’s eyes and the frequencies for each. Next, use the table to find the relative frequencies for each number of bull’s eyes. The relative frequencies are the probability distribution. Probability Distribution of Bull’s Eye’s Hits Number of bull’s eyes Hit 0 1 2 3 4 5 6 Frequency 3 5 7 7 11 10 7 Probability 3/50 1/10 7/50 7/50 11/50 1/5 7/50 Your Turn On a math test, there were 10 scores between 90 and 100, 12 scores between 80 and 89, 15 scores between 70 and 79, 8 scores between 60 and 69, and 2 scores below 60. What is the probability distribution for the test scores? Probability Distribution of Bull’s Eyes Hits Scores Ranges 90-100 80-89 70-79 60-69 Below 60 Frequency 10 12 15 8 2 Probability 10/47 12/47 15/47 8/47 2/47 Permutations and Combinations • Fundamental Counting Principle says that if event M occurs in m ways and event N occurs in n ways, then event M followed by event N can occur in m x n ways. • Permutation – an arrangement of items in which the order of the objects is important. • n factorial – The factorial of a number is the product of the natural numbers less than or equal to the number. • Combination – an arrangement of items in which the order is not important. Fundamental Counting Principle You have a red shirt, a blue shirt, a pair of black pants and a pair of khaki pants. How many different outfits can you make. Red shirt and black pants Red shirt and khaki pants Blue shirt and black pants Blue shirt and khaki pants Total of 4 different outfits 2 shirts times 2 pairs of pants = 4 different outfits. Fundamental Counting Principle The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. In how many ways can a person order a two-course meal? Solution: Choosing from one of 6 appetizers and one of 14 main courses, the total number of two-course meals is 6× 14 = 84 Your Turn A restaurant offers 10 appetizers and 15 main courses. In how many ways can you order a twocourse meal? Solution: 10× 15 = 150 two-course meals The Fundamental Counting Principle with More Than Two Groups of Items Next semester you are planning to take three courses – math, English and humanities. Based on time blocks and highly recommended professors, there are 8 sections of math, 5 of English, and 4 of humanities that you find suitable. Assuming no scheduling conflicts, how many different threecourse schedules are possible? Solution: math English humanities 8 ×5 × 4 = 160 Your Turn A pizza can be ordered with two choices of size(medium or large), three choices of crust (thin, thick, or regular), and five choices of toppings ( ground beef, sausage, pepperoni, bacon, or mushrooms.) How many different one-topping pizzas can be ordered? Solutions: 2× 3 × 5 = 30 Permutations Order Matter One item can be arranged one way: 1 permutation A Two items can be arranged two ways: 2 x 1 permutations AB and BA Three items can be arranged six ways: ABC, ACB, BAC, BCA, CAB, CBA 3 x 2 x1 permutations n Factorial The factorial of a number is the product of the natural numbers less than or equal to the number. 0! Is defined as 1 6!=6 x 5 x4 x3 x2 x 1=720 n!=n x (n-1) x (n-2) x (n-3) x … x 1 Finding the Number of Permutations You download 8 songs on you music player. If you play the songs using the random shuffle option, how many different ways can the sequence of songs be played? 8!=8x7x6x5x4x3x2x1=40,320 Your Turn In how many ways can you arrange 12 books on a shelf? 12!=12x11x10x9x8x7x6x5x4x3x2x1=479,001,600 Sometimes you may not want to order an entire set of items. Suppose you want to select and order 3 people from a group of 7. One way to find possible permutations is to use the Fundamental Counting Principle. First person Second person Third person 7 choices x 6 choices x 5 choices =210 permutations. Another way to find the possible permutations is to use factorials. You can divide the total number of arrangements by the number of arrangements that are not used. 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 7 𝑝𝑒𝑜𝑝𝑙𝑒 7! 7𝑥6𝑥5𝑥4𝑥3𝑥2𝑥1 = = 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 4 𝑝𝑒𝑜𝑝𝑙𝑒 4! 4𝑥3𝑥2𝑥1 = 210 Notice: 4x3x2x1 will cancel so 7x6x5=210 Permutation Formula The number of permutations of n items taken r at a time. nPr= 𝑛! 𝑛−𝑟 ! The number of permutation of 7 items taken 3 at a time. 7P3= 7! 7! = 7−3 ! 4! Finding a Permutation How many ways can a club select a president, a vice president, and a secretary from a group of 5 people? n=5 since there are 5 people to choose from. r=3 since there are 3 officer positions. 5P3= 5! 5−3 ! = 5! 2! = 5𝑥4𝑥3𝑥2𝑥1 2𝑥1 = 5x4x3 = 60 Your Turn An art gallery has 9 fine-art photographs from an artist and will display 4 from left to right along a wall. In how many ways can the gallery select and display the 4 photographs? 9P4= 9! 9−4 ! = 3024 Combinations Formula To find the number of combinations, the formula for permutations can be modified. Number of 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑎𝑟𝑟𝑎𝑛𝑔𝑒 𝑎𝑙𝑙 𝑖𝑡𝑒𝑚𝑠 permutations= 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑎𝑟𝑟𝑎𝑛𝑡 𝑖𝑡𝑒𝑚𝑠 𝑛𝑜𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 Because order does not matter, divide the number of permutations by the number of ways to arrange the selected items. Number of 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑎𝑟𝑟𝑎𝑛𝑔𝑒 𝑎𝑙𝑙 𝑖𝑡𝑒𝑚𝑠 combinations=(𝑤𝑎𝑦𝑠 𝑡𝑜 𝑎𝑟𝑟𝑎𝑛𝑔𝑒 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑡𝑒𝑚𝑠)(𝑤𝑎𝑦𝑠 𝑡𝑜 𝑎𝑟𝑟𝑎𝑛𝑔𝑒 𝑖𝑡𝑒𝑚𝑠 𝑛𝑜𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑) Combinations Formula The number of combinations of n items taken r at a time is 𝑛! nCr=𝑟! 𝑛−𝑟 ! The number of combinations of 7 items taken 3 at a time is 7! 7C3=3! 7−3 ! Using the Combination Formula Katie is going to adopt kittens from a litter of 11. How many ways can she choose a group of 3 kittens? Step 1: Is this a permutation or a combination? This is a combination since order doesn’t matter. Kitty, Smoky and Tiger is the same as Tiger, Kitty and Smoky. Step 2: Use the formula 11! 11C3=3! 11−3 ! = 11𝑥10𝑥9𝑥8𝑥7𝑥65𝑥4𝑥3𝑥2𝑥1 3𝑥2𝑥1𝑥(8𝑥7𝑥6𝑥5𝑥4𝑥3𝑥2𝑥1) = 11𝑥10𝑥9 3𝑥2𝑥1 = 165 Your Turn The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected? 8! 8C2=2! 8−2 ! = 28 Identifying combinations and Permutations To determine whether to use the permutation formula or the combination formula, you must decide whether order is important. A. A college student is choosing 3 classes to take during first, second and third semester from the 5 elective classes offered in his major. How many possible ways can the student schedule the three classes. Since the order does matter this is a permutation. Identifying Combinations and Permutations B. A jury of 12 people is chosen from a pool of 35 potential jurors. How many different juries can be chosen? Since order doesn’t matter this is a combination. Your Turn A yogurt shop allows you to choose any 3 of the 10 possible mix-ins for a Just Right Smoothie. How many different Just Right Smoothies are possible? Since order doesn’t matter this is a combination. Compound Probability • Compound event – an event that is made up of two or more events • Independent events – an event that does not affect how another event occurs. • Dependent events – an event that does affect how another event occurs. • Mutually exclusive events – events that cannot happen at the same time • Overlapping events – have common outcomes Identifying Independent and Dependent Events Are the outcomes of each trial independent or dependent events? A. Choose a number tile from 12 tiles. Then spin a spinner. The choice of number tile does not affect the spinner result. The events are independent. B. Pick on card from a set of 15 sequentially numbered cards. Then, without replacing the card, pick another card. The first card chosen affects the possible outcomes of the second pick, so the events are dependent. Your Turn You roll a standard number cube. Then you flip a coin. Are the outcomes independent or dependent events. Explain. They are independent because the roll of a die does not affect the flip of a coin. Independent Events If A and B are independent event, then P(A and B)=P(A) x P(B) Finding the Probability of Independent Events Find each probability. A. Spinning 4 then 4 again on the spinner. P(4 and then 4)=P(4) x 1 1 1 P(4)=( ) = 6 6 36 B. Spinning 5 then 3. P(5 and then 3)=P(5) x 1 1 1 P(3)=( ) = 3 4 12 Your Turn Find each probability. A. Rolling a 6 on one number cube and a 6 on another. P(6 on one cube and 6 on another)=P(6) x P(6)= 1 1 1 ( ) = 6 6 36 B. Tossing heads, then heads and tails when tossing a coin 3 times. P(tossing head, then heads and then tails)=P(heads) x 1 1 1 1 P(heads) x P(tails)=( ) = 2 2 2 8 Dependent Events To find the probability of dependent events you can use conditional probability P( 𝐵 𝐴 If A and B are dependent events, then P(A and B)=P(A) x P( 𝐵 𝐴 , where P( 𝐵 𝐴 is the probability of B given that A has occurred. Finding the Probability of Dependent Events Two number cubes are rolled – one red and one blue. Explain why the events are dependent. Then find the indicated probability. A. The red cube shows a 1 and the sum is less than 4. Why the events are dependent. The events “the red cube shows a 1” and “the sum is less than 4” are dependent because P(sum<4) is different when it is known that a red 1 has occurred. 1 6 P(red 1)= 1 3 P( 𝑠𝑢𝑚 < 4 𝑟𝑒𝑑 1 = only 1 and 2 will meet the condition. P(red1 and sum<4)=P(red 1) x P( 𝑠𝑢𝑚 < 4 𝑟𝑒𝑑 1 = 1 1 ( ) 6 3 = 1 18 Finding the Probability of Dependent Events Two number cubes are rolled – one red and one blue. Explain why the events are dependent. Then find the indicated probability. B. The blue cube sows a multiple of 3 and the sum is 8. The events are dependent because P(sum is 8) is different when the blue cube shows a multiple of 3. P(blue multiple of 3)= 1 3 2 1 P 𝑠𝑢𝑚 𝑖𝑠 8 𝑏𝑙𝑢𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3 = = (3+5 , 6+2) 12 6 P(blue multiple of 3 and sum is 8)= P(blue multiple of 3) x 1 1 1 P 𝑠𝑢𝑚 𝑖𝑠 8 𝑏𝑙𝑢𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3 = = 3 6 18 Your Turn Two number cubes are rolled – one red and one black. Explain why the events are dependent, and then find the indicated probability. The red cube shows a number greater than 4 and the sum is greater than 9. P(sum of 9) changes when it is know that the red cube is >4. 2 6 P(>4)= = 1 3 5 x P 𝑠𝑢𝑚 > 9 > 4 = five of the possible 12 outcomes are >9. (5+5, 12 5+6, 6+4, 6+5, 6+6) 1 3 P(>4 and sum >9)=P(>4) x P 𝑠𝑢𝑚 > 9 > 4 =( )( 5 12) = 5 36 Mutually Exclusive Events If A and B are mutually exclusive events, then P(A and B)=0, and P(A or B)=P(A) +P(B). Finding the Probability of Mutually Exclusive Events Student athletes at a local high school may participate in only one sport each season. During the fall season, 28% of student athletes play basketball and 24% are on the swim team. What is the probability that a randomly selected student athlete plays basketball or is on the swim team? Because athletes participate in only one sport each season, the events are mutually exclusive. Use the formula P(A or B)= P(A) + P(B) P(basketball or swim team) = P(basketball) + P(swim team)=28%+24%=52% Your Turn In the Spring season, 15% of the athletes play baseball and 23% are on the track team. What is the probability of an athlete either playing baseball or being on the track team? P(baseball or track team)=P(baseball)+P(track team)=15%+23%=38% Probability of Overlapping Events If A and B are overlapping events, the P(A or B)=P(A) + P(B) – P(A and B) Suppose you have 7 index cards, each having one of the following letters written on it: A B C D E F G P(FACE), the probability of selecting a letter from the word FACE, is 4/7. P(CAB), the probability of selecting a letter from the word CAB, is 3/7 D F E G A C B Consider P(FACE or CAB), the probability of choosing a letter from either the word FACE or the word CAB. These events overlap since the words have two letters in common. If you simply add P(FACE) and P(CAB), you get 4/7+3/7=7/7. The value of the numerator should be the number of favorable outcomes, but there are only 5 distinct letters in the words FACE and CAB. The problem is that when you simply add, the letters A and C are counted twice, once in the favorable outcomes for the word FACE, and once for the favorable out comes for the word CAB. You must subtract the number of letters that the two words have in common so they are only counted once. P(FACE or CAB)=(4+5-2)/7=4/7+5/7-2/7=P(FACE)+P(CAB)-P(AC) Find Probabilities of Overlapping Events What is the probability of rolling either an even number or a multiple of 3 when rolling a standard number cube? Know Need You are rolling a standard number cube. The events are overlapping events because 6 is both even and a multiple of 3. You need the probability of rolling an even number and the probability of rolling a multiple of 3 Plan Find the probabilities and use the formula for probabilities of overlapping events. P(even or multiple of 3)=P(even) +P(multiple of 3)-P(even and multiple of 3)=3/6+2/61/6=4/6=2/3 Your Turn What is the probability of rolling either an odd number or a number less than 4 when rolling a standard number cube? P(odd or <4)=P(odd)+P(<4)-P(odd and <4)=3/6+3/6-2/6=4/6=2/3 Probability Models • Two-way frequency table – displays the frequencies of data in two different categories. • Conditional probability – the probability that an event will occur, given that another event has already occurred Using a Two-Way Frequency Table The table shows data about student involvement in extracurricular activities at a local high school. What is the probability that a randomly chosen student is a female who is not involved in extracurricular activities? Involved in Activities Not Involved in Activities Totals Male 112 145 257 Female 139 120 259 Total 251 265 516 To find the probability, calculate the relative frequency. Relative 0.233 𝑓𝑒𝑚𝑎𝑙𝑒𝑠 𝑛𝑜𝑡 𝑖𝑛𝑣𝑜𝑙𝑣𝑒𝑑 frequency= 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 = 120 516 ≈ Your Turn The two-way frequency table at the right shows the number of male and female students by grade level on the prom committee. What is the probability that a member of the prom committee is a male who is a junior? Male Female Totals Juniors 3 4 7 Seniors 3 2 5 Totals 6 6 12 𝑗𝑢𝑛𝑖𝑜𝑟 𝑚𝑎𝑙𝑒 3 Relative frequency=𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑟𝑜𝑚 𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑒𝑒 = 12 = .25 Finding Probability Respondents of a poll were asked whether they were for, against, or had no opinion about a bill before the state legislature that would increase the minimum wage. What is the probability that a randomly selected person is over 60 years old, given that the person had no opinion on the state bill? Age For Group Against No opinion Totals 18-29 310 50 20 380 30-45 200 30 10 240 46-60 120 20 30 170 Over 60 150 20 40 210 Totals 780 120 100 1000 The condition that the person selected has no opinion on the minimum-wage bill limits the total outcomes to the 100 people who had no opinion. Of those 100 people, 40 respondents were over 60 years old. 40 P 𝑜𝑣𝑒𝑟 60 𝑛𝑜 𝑜𝑝𝑖𝑛𝑖𝑜𝑛 = 100=.4 Your Turn Respondents of a poll were asked whether they were for, against, or had no opinion about a bill before the state legislature that would increase the minimum wage. What is the probability that a randomly selected person is 30-45 years old, given that the person is in favor of the minimum-wage bill? Age For Group Against No opinion Totals 18-29 310 50 20 380 30-45 200 30 10 240 46-60 120 20 30 170 Over 60 150 20 40 210 Totals 780 120 100 1000 200 P 30 − 45 𝐹𝑜𝑟 = 780 ≈ .256 Using Relative Frequencies A company has 150 sales representatives. Two months after a sales seminar, the company vicepresident made the table based on sales results. What is the probability that someone who attended the seminar had an increase in sales? Attended Did not Totals Seminar Attend Seminar Increased Sales .48 .02 .5 No Increase in Sales .32 .18 .5 Totals .8 .2 1 P 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑠𝑎𝑙𝑒𝑠 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 = 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑎𝑡𝑡𝑒𝑛𝑑 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 𝑎𝑛𝑑 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑎𝑡𝑡𝑒𝑛𝑑𝑒𝑑 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 P 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑠𝑎𝑙𝑒𝑠 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 = .48 .8 = .6 Your Turn A company has 150 sales representatives. Two months after a sales seminar, the company vicepresident made the table based on sales results. What is the probability that a randomly selected sales representative, who did not attend the seminar, did not see an increase in sales? Attended Did not Totals Seminar Attend Seminar Increased Sales .48 .02 .5 No Increase in Sales .32 .18 .5 Totals .8 .2 1 P 𝑛𝑜 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑛𝑜 𝑠𝑎𝑙𝑒𝑠 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 = 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑛𝑜𝑡 𝑎𝑡𝑡𝑒𝑛𝑑 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 𝑎𝑛𝑑 𝑛𝑜 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑑𝑖𝑑 𝑛𝑜𝑡 𝑎𝑡𝑡𝑒𝑛𝑑 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 P 𝑛𝑜 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑛𝑜 𝑠𝑎𝑙𝑒𝑠 𝑠𝑒𝑚𝑖𝑛𝑎𝑟 = .18 .5 = .36 Conditional Probability Formulas For any two events A and B, the probability of B occurring, given that event A has occurred, is P 𝐵𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) , 𝑤ℎ𝑒𝑟𝑒 𝑃(𝐴) 𝑃(𝐴) ≠ 0 Using Conditional Probabilities In a study designed to test the effectiveness of a new drug, half of the volunteers received the drug. The other half of the volunteers received a placebo, a tablet or pill containing no medication. The probability of a volunteer receiving the drug and getting well was 45%. What is the probability of someone getting well, given that he receives the drug? Step 1: Identify the probabilities P 𝐵 𝐴 = 𝑃(𝑔𝑒𝑡𝑡𝑖𝑛𝑔 𝑤𝑒𝑒, 𝑔𝑖𝑣𝑒𝑛 𝑡𝑎𝑘𝑖𝑛𝑔 𝑡ℎ𝑒 𝑛𝑒𝑤 𝑑𝑟𝑢𝑔) P(A)=P(taking the new drug)=1/2 =.5 P(A and B)=P(taking the new drug and getting well)=45% or .45 Step 2: Find P 𝐵 𝐴 P 𝐵𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴) = .45 .5 = .9 Your Turn The probability of a volunteer receiving the placebo and having his or her health improve was 20%. What is the conditional probability of a volunteer’s health improving, given that they received the placebo? P 𝐵 𝐴 =P(getting well, given the placebo) P(A)= P(taking the placebo)=.5 P(A and B)=(taking the placebo and getting well)=.2 P 𝐵 𝐴 =.2/.5=.4 Comparing Conditional Probabilities In a survey of pet owners, 45% own a dog, 27% own a cat and 12% own both a dog and a cat. What is the conditional probability that a dog owner also owns a cat? What is the conditional probability that a cat owner also owns a dog? P 𝑐𝑎𝑡 𝑑𝑜𝑔 = .12 .45 = .267 P 𝑑𝑜𝑔 𝑐𝑎𝑡 = .12 .27 = .444 Your Turn The same survey showed that 5% of the pet owners own a dog, a cat, and at least one other type of pet. a. What is the conditional probability that a pet owner owns a cat and some other type of pet, given that they own a dog? .05/.45=.111 b. What is the conditional probability that a pet owner owns a dog and some other type of pet, given that they own a cat? .05/.27=.185 Modeling Randomness You can use probability to make choices and to help make decisions based on prior experience. A random event has no predetermined pattern or bias toward one outcome or another. You can use random number tables or randomly generated numbers using graphing calculators or computer software to help you make fair decisions. We will be using a random number table. Making Random Selections There are 28 students in a homeroom. Four students are chosen at random to represent the homeroom on a student committee. How can a random number table be used to fairly choose the students? Step 1: Select a line from a random number table. 18823 18160 93593 67294 09632 62617 86779 Step 2: Group the line from the table into two digit numbers. 18 82 31 81 60 93 59 36 72 94 09 63 26 26 17 86 77 9 Step 3: Match the first four numbers less than 28 with the position of the students’ names on a list. Duplicates and numbers greater than 28 are discarded because they don’t correspond to any student on the list. 18 09 26 17 The students listed 18th , 9th, 26th, and 17th on the list are chosen fairly. Your Turn A teach wishes to choose three students from a class of 25 students to raise the school’s flag. What numbers should the teacher choose based on the line from a random number table? 65358 70469 87149 89509 72176 18103 55169 799954 72002 20582 65 35 87 04 69 87 14 98 95 09 72 17 61 81 03 55 16 97 99 95 47 20 02 20 58 2 04, 14, 09, The students listed 4th, 14th, and 9th on the list are chosen fairly