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1 Probability Probability: the measure of the likelihood of an event. All probabilities must be 0 < P < 1, with P(E)= 0 meaning that the event cannot happen (impossibility). P(E) = 1, means that the event will definitely occur (certainty). Probabilities can be expressed as 0, 1, fractions or decimals. Theoretical Probability of an event: P(E) = number of favorable outcomes total number of possible outcomes Example: A standard die has six sides with numbers 1-6 on it. The probability of rolling a 5: P(5) = 1/6, because there is 1 side with a 5 on it out of 6 possible sides. The probability of an event not happening [P(not E)] is calculated by 1 – P(E) Example: The probability that it will rain is 5/9. The probability that it will not rain is (1 – 5/9) = 4/9. Using a tree diagram to show a sample space: Tree Diagram: A visual method of displaying all the outcomes in a sample space by using “branches” that resemble a tree. Example: Bobby has 4 shirts (white, blue, red, and yellow). He also has 3 ties (blue, red, and yellow). Each morning he selects a shirt and a tie at random. Make a tree diagram for all the possible combinations of shirt and tie. Then use the tree diagram to find the probability of: 1) A blue shirt 2) A red tie 3) Selecting a yellow shirt and yellow tie. 4) Selecting a yellow shirt or a yellow tie. (Keep in mind the difference between “and” and “or”.) SHIRTS TIES 2 Probability 1) A deli offers sandwiches with three choices of bread (white, wheat, or rye). There are also three choices for meat (ham, roast beef, and turkey), and a choice of mayo or no mayo. Make a tree diagram that represents all the possible combinations of bread, meat, and mayo. Then use the tree diagram to answer the following questions. a. P(white bread and no mayo) b. P(not wheat) c. P(roast beef or mayo) d. P(tuna) 2) The Jeep dealer offers the Grand Cherokee in four colors (black, white, silver, and yellow). There is a choice of tinted windows or non-tinted windows. There is also a choice between standard transmission and automatic. Make a tree diagram to represent the sample space of all the possible combinations. Find: a. P(black) b. P(not white) c. P(yellow or tinted windows) d. P(silver and tinted windows) e. P(standard or automatic) f. P(standard and automatic) 3) At August Madness the cafeteria will be offering a box lunch that consists of a sandwich, bag of chips, and a drink. There are 4 different sandwiches (cheese, chicken, turkey, and peanut butter). There are 4 different types of chips(Doritos, pretzels, potato chips, and fritos). There are 3 different drinks (chocolate milk, water, and orange juice). Make a tree diagram to represent the sample space of all the possible combinations. Find: a. P(chocolate milk and water) b. P(peanut butter or fritos) c. P(turkey or chicken) d. P (cheese) e. P(potato chips and water) 4) Janice picks out an SFP dress code approved outfit at random everyday. She picks a blue or gray skirt. Then she picks a red, white, or navy polo shirt. Janice then picks either navy or gray stockings. And finally Janice picks either brown or black shoes. Make a tree diagram to represent the sample space of all the possible combinations. Find: a. P(gray stockings and white polo) b. P(green skirt) c. P(black shoes or brown shoes) d. P(blue skirt or red polo) e. P(red polo and a white polo) 3 Histograms 1) Jason played in 20 basketball games. The amount of points he scored in each game is listed below. a) Complete the frequency table and construct a frequency histogram. b) Which interval contains the median? c) What is the mode? 15 41 24 31 26 27 26 35 Interval 0-9 10-19 20-29 30-39 49-49 11 14 8 38 Tally 21 25 36 29 Frequency 17 32 41 38 4 2) A class of 24 students took the SAT. Their scores are listed below. a) Complete the frequency table and construct a frequency histogram. b) Which interval contains the median? c) What is the mode? 1120 710 970 770 Interval 700-799 800-899 900-999 1000-1099 1100-1199 1200-1299 1300-1399 1400-1499 1500-1599 1510 1130 990 1120 740 890 860 1130 Tally 1170 910 1310 1080 Frequency 890 1020 1290 1280 1250 1050 1030 1010 5 3) There are 28 students in a science class. Their grades from their most recent exam are listed below. a) Complete the frequency table and construct a frequency histogram. b) Which interval contains the median? c) What is the mode? 91 87 78 76 72 58 92 78 Interval 51-60 61-70 71-80 81-90 91-100 93 84 52 98 Tally 100 78 86 95 Frequency 57 99 94 83 72 93 100 75 85 91 89 85 6 4) There are 36 students in cor 117. The number of times each student was absent is listed below. a) Complete the frequency table and construct a frequency histogram. b) Which interval contains the median? c) What is the mode? 3 0 10 3 13 4 6 5 6 2 0 6 Interval 0-2 3-5 6-8 9-11 12-14 0 7 4 6 7 0 Tally 0 3 0 5 2 0 Frequency 4 1 0 4 5 2 11 4 5 9 3 1 7 5) There are 24 students in a math class. Their grades from their most recent exam are listed below. a) Complete the frequency table and construct a frequency histogram. b) Complete the cumulative frequency table and construct a cumulative freq. hist. c) Which interval contains the median? d) What is the mode? e) What interval contains the lower quartile? f) What interval contains the UQ? 92 72 92 82 57 79 87 58 84 72 92 93 78 84 65 86 88 88 92 92 85 89 83 75 Range Tally Frequency Range 51-60 51-60 61-70 51-70 71-80 51-80 81-90 51-90 91-100 51-100 Cumulative Frequency 8 6) Javier was the starting pitcher in 28 baseball games. The amount of points he scored in each game is listed below. a) Complete the frequency table and construct a frequency histogram. b) Which interval contains the median? c) What is the mode? d) Which interval contains the lower quartile? e) Which contains the UQ? 6 8 2 0 6 7 8 9 7 6 3 6 4 3 5 3 2 1 9 6 2 9 6 5 4 5 6 3 Range Tally Frequency Range 0-1 0-1 2-3 0-3 4-5 0-5 6-7 0-7 8-9 0-9 Cumulative Frequency