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Probability

Probability. Foldable #1 Do a hotdog bun fold and make 10 cuts . Vocabulary Words. C ombination - a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter.

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Probability

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  1. Probability

  2. Foldable #1Do a hotdog bun fold and make 10 cuts

  3. 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.

  4. 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= • 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.

  5. 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 reasoningP(event)=

  6. 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. P(event)=

  7. Experimental Probability Experimental probability of an event – the number of times that the event occurs. experimental probability=

  8. Finding Experimental Probability The table shows the results of a spinner experiment. Find each experimental probability. A. Spinning a 4 =28% There were 50 occurrences B. Spinning a number greater than 2

  9. Theoretical Probability • Theoretical probability – the ratio of the number of favorable outcomes to the total number of outcomes. P(event)=

  10. 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 song)=

  11. 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 = • Probability distribution – shows the probability of each possible outcome.

  12. 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

  13. 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? =

  14. 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?

  15. 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=

  16. 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? 18/100=9/50

  17. 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?

  18. Finding a Probability Distribution 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.

  19. 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?

  20. 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.

  21. 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.

  22. 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

  23. Your Turn A restaurant offers 10 appetizers and 15 main courses. In how many ways can you order a two-course meal? Solution: 10two-course meals

  24. 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 three-course schedules are possible? Solution: math English humanities 8

  25. 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

  26. PermutationsOrder Matter One item can be arranged one way: A 1 permutation Two items can be arranged two ways: AB and BA 2 x 1 permutations Three items can be arranged six ways: ABC, ACB, BAC, BCA, CAB, CBA 3 x 2 x1 permutations

  27. 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

  28. 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

  29. Your Turn In how many ways can you arrange 12 books on a shelf? 12!=12x11x10x9x8x7x6x5x4x3x2x1=479,001,600

  30. 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.

  31. 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. Notice: 4x3x2x1 will cancel so 7x6x5=210

  32. Permutation Formula The number of permutations of n items taken r at a time. nPr= The number of permutation of 7 items taken 3 at a time. 7P3==

  33. 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=

  34. 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=

  35. Combinations Formula To find the number of combinations, the formula for permutations can be modified. Number of permutations= Because order does not matter, divide the number of permutations by the number of ways to arrange the selected items. Number of combinations=

  36. Combinations Formula The number of combinations of n items taken r at a time is nCr= The number of combinations of 7 items taken 3 at a time is 7C3=

  37. 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 11C3=

  38. 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? 8C2=

  39. Identifying combinations and Permutations To determine whether to use the permutation formula or the combination formula, you must decide whether order is important. • 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.

  40. 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.

  41. 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.

  42. 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

  43. Identifying Independent and Dependent Events Are the outcomes of each trial independent or dependent events? • 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.

  44. 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.

  45. Independent Events If A and B are independent event, then P(A and B)=P(A) x P(B)

  46. Finding the Probability of Independent Events Find each probability. • Spinning 4 then 4 again on the spinner. P(4 and then 4)=P(4) x P(4)=( B. Spinning 5 then 3. P(5 and then 3)=P(5) x P(3)=(

  47. Your Turn Find each probability. • 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)= ( B. Tossing heads, then heads and tails when tossing a coin 3 times. P(tossing head, then heads and then tails)=P(heads) x P(heads) x P(tails)=(

  48. Dependent Events To find the probability of dependent events you can use conditional probability P( If A and B are dependent events, then P(A and B)=P(A) x P(, where P( is the probability of B given that A has occurred.

  49. 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. • 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. P(red 1)= P(only 1 and 2 will meet the condition. P(red1 and sum<4)=P(red 1) x P( = (

  50. 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)= P= (3+5 , 6+2) P(blue multiple of 3 and sum is 8)= P(blue multiple of 3) x P

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