1. Bennie D Waller, Longwood University Probability

2. Bennie D Waller, Longwood University Probability  Probability - the likelihood of an event occurring  0<=P(X)<=1

3. Bennie D Waller, Longwood University Probability  Classical Probability –flipping a coin  Empirical Probability – based on prior events – airline arrival  Subjective Probability – Longwood baseball winning CWS

4. Bennie D Waller, Longwood University Probability  Marginal Probability – the likelihood of a particular event occurring P(Smoker)  Joint Probability – the likelihood of two events both occurring – P(Male AND Smoker)  Conditional Probability – likelihood of an event occurring based the previous outcome of another event – P(Rain | Cloudy)  Independence – two events are independent if one’s outcome does not impact the other’s outcome  Mutually exclusive – if one event occurs, the other cannot

5. Bennie D Waller, Longwood University Probability P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = P(A) + P(B) Rules of addition

6. Bennie D Waller, Longwood University P(A and B) = P(A)P(B|A) P(A and B) = P(A)P(B) Probability Rules of multiplication

7. Bennie D Waller, Longwood University Contingency Tables.30.05.20.45 MaleFemale Smoker Non-smoker.50.35.65

8. Bennie D Waller, Longwood University Examples Problem: A student is taking two courses, history and math. The probability that the student will pass the history class is.60 and the probability of passing the math class is.70. The probability of passing both is.50. What is the probability of passing at least one of the classes?

9. Bennie D Waller, Longwood University Problem: A study by the National Park Service revealed that 50% of the vacationers going to the Rocky Mountain region visit Yellowstone Park, 40% visit the Tetons and 35% visit both. 1.What is the probability that a vacationer will visit at least one of these magnificent attractions? 2.What is the.35 probability called? 3.Are these events mutually exclusive?.50.40.35

10. Bennie D Waller, Longwood University General Multiplication Rule A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry. What is the likelihood both shirts selected are white? P(A and B) = P(A)*P(B|A)

11. Bennie D Waller, Longwood University Contingency Tables In a survey of employee satisfaction, the following table summarizes the results in terms of employee satisfaction and gender. 1.What is the probability that an employee is Female and Dissatisfied? 2.What is the probability that an employee is Male or Dissatisfied? 3.What is the probability that an employee is Satisfied given that the employee is Male?

12. Bennie D Waller, Longwood University Problem: Airlines monitor the causes of flights arriving late. 75% of flights are late because of weather, 35% of flights are late because of ground operations. 15% of flights are late because of weather and ground operations. What is the probability that a flight arrives late because of weather or ground operations?

13. Bennie D Waller, Longwood University Discrete Probability Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

14. Bennie D Waller, Longwood University Discrete Probability Distributions DISCRETE RANDOM VARIABLE - A variable that can assume only certain clearly separated values and is the typically the result of counting something. CHARACTERISTICS OF A PROBABILITY DISTRIBUTION 1.The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive events. 3. The list is exhaustive. So the sum of the probabilities of the various events is equal to 1. CONTINUOUS RANDOM VARIABLE – A variable that can assume an infinite number of values within a given range. It is usually the result of some type of measurement

15. Bennie D Waller, Longwood University Experiment: Toss a coin three times. Observe the number of heads. The possible results are: Zero heads, One head, Two heads, and Three heads. What is the probability distribution for the number of heads? PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome. 6-15 Probability Distributions

16. Bennie D Waller, Longwood University Discrete Probability Distribution Mean XP(x).05.20.06.20.07.20.08.20.09.20

17. Bennie D Waller, Longwood University Discrete Probability Distribution Variance XP(x).05.20.06.20.07.20.08.20.09.20

18. Bennie D Waller, Longwood University Example Problem: For the following probability distribution, what is the variance?

19. Bennie D Waller, Longwood University Continuous Probability Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

20. Bennie D Waller, Longwood University Characteristics of a Normal Probability Distribution 1.It is bell-shaped. 2.It is symmetrical about the mean 3.It is asymptotic: 4.The location of a normal distribution is determined by the mean, , the dispersion is determined by the standard deviation,σ. 5.The arithmetic mean, median, and mode are equal 6.The total area under the curve is 1.00; half the area under the normal curve is to the right of this center point, the mean, and the other half to the left of it. 7-20 Continuous Distributions

21. Bennie D Waller, Longwood University Converting to Standard Normal Distribution

22. Bennie D Waller, Longwood University Converting to Standard Normal Distribution

23. Bennie D Waller, Longwood University The Standard Normal Probability Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z distribution. A z-value is the signed distance between a selected value, designated X, and the population mean , divided by the population standard deviation, σ. The formula is: 7-23 Converting to Standard Normal Distribution

24. Bennie D Waller, Longwood University Standard Normal Distribution Tables z0.000.010.020.030.040.050.060.070.080.09 0.00.00000.00400.00800.01200.01600.01900.02390.02790.03190.0359 0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879 0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224 0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549 0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852 0.80.28810.29100.29390.29690.29950.30230.30510.30780.31060.3133 0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.00.34130.34380.34610.34850.35080.35130.35540.35770.35290.3621 1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857 2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890 2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916 2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936 2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952 2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964 2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974 2.80.49740.49750.49760.4977 0.49780.4979 0.49800.4981 2.90.49810.4982 0.49830.4984 0.4985 0.4986 3.00.4987 0.4988 0.4989 0.4990 3.10.49900.4991 0.4992 0.4993 3.20.4993 0.4994 0.4995 3.30.4995 0.4996 0.4997 3.40.4997 0.4998 Areas Under the Normal Curve

25. Bennie D Waller, Longwood University Standard Normal Distribution Tables z0.000.010.020.030.040.050.060.070.080.09 0.00.00000.00400.00800.01200.01600.01900.02390.02790.03190.0359 0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879 0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224 0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549 0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852 0.80.28810.29100.29390.29690.29950.30230.30510.30780.31060.3133 0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.00.34130.34380.34610.34850.35080.35130.35540.35770.35290.3621 1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857 2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890 2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916 2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936 2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952 2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964 2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974 2.80.49740.49750.49760.4977 0.49780.4979 0.49800.4981 2.90.49810.4982 0.49830.4984 0.4985 0.4986 3.00.4987 0.4988 0.4989 0.4990 3.10.49900.4991 0.4992 0.4993 3.20.4993 0.4994 0.4995 3.30.4995 0.4996 0.4997 3.40.4997 0.4998

26. Bennie D Waller, Longwood University Problem: The weight of a bag of corn chips is normally distributed with a mean of 22 ounces and a standard deviation of ½ ounces. What is the probability that a bag of corn chips weighs more than 23 ounces? Tables z0.000.010.020.030.040.050.060.070.080.09 0.00.00000.00400.00800.01200.01600.01900.02390.02790.03190.0359 0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879 0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224 0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549 0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852 0.80.28810.29100.29390.29690.29950.30230.30510.30780.31060.3133 0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.00.34130.34380.34610.34850.35080.35130.35540.35770.35290.3621 1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857 2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890 2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916 2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936 2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952 2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964 2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974 2.80.49740.49750.49760.4977 0.49780.4979 0.49800.4981 2.90.49810.4982 0.49830.4984 0.4985 0.4986 3.00.4987 0.4988 0.4989 0.4990 3.10.49900.4991 0.4992 0.4993 3.20.4993 0.4994 0.4995 3.30.4995 0.4996 0.4997 3.40.4997 0.4998

27. Bennie D Waller, Longwood University Problem: A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $30,000 and a standard deviation of $3,000. How many students earned between $27,000 and $33,000? Tables z0.000.010.020.030.040.050.060.070.080.09 0.00.00000.00400.00800.01200.01600.01900.02390.02790.03190.0359 0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879 0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224 0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549 0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852 0.80.28810.29100.29390.29690.29950.30230.30510.30780.31060.3133 0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.00.34130.34380.34610.34850.35080.35130.35540.35770.35290.3621 1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857 2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890 2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916 2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936 2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952 2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964 2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974 2.80.49740.49750.49760.4977 0.49780.4979 0.49800.4981 2.90.49810.4982 0.49830.4984 0.4985 0.4986 3.00.4987 0.4988 0.4989 0.4990 3.10.49900.4991 0.4992 0.4993 3.20.4993 0.4994 0.4995 3.30.4995 0.4996 0.4997 3.40.4997 0.4998

28. Bennie D Waller, Longwood University Problem: The weight of a bag of corn chips is normally distributed with a mean of 22 ounces and a standard deviation of ½ ounces. What is the probability that a bag of corn chips weighs between 20.75 and 23.25 ounces? Tables z0.000.010.020.030.040.050.060.070.080.09 0.00.00000.00400.00800.01200.01600.01900.02390.02790.03190.0359 0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879 0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224 0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549 0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852 0.80.28810.29100.29390.29690.29950.30230.30510.30780.31060.3133 0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.00.34130.34380.34610.34850.35080.35130.35540.35770.35290.3621 1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857 2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890 2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916 2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936 2.5 0.4938 0.49400.49410.49430.49450.49460.49480.49490.49510.4952 2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964 2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974 2.80.49740.49750.49760.4977 0.49780.4979 0.49800.4981 2.90.49810.4982 0.49830.4984 0.4985 0.4986 3.00.4987 0.4988 0.4989 0.4990 3.10.49900.4991 0.4992 0.4993 3.20.4993 0.4994 0.4995 3.30.4995 0.4996 0.4997 3.40.4997 0.4998

29. Bennie D Waller, Longwood University End