Probability and Statistics1  Basic Probability  Binomial Distribution  Statistical Measures  Normal Distribution

FE Reference Handbook  Published by the National Council of Examiners for Engineering and Surveying (NCEES)  Available electronically at exam  Only reference material allowed at exam  Free preview copy (PDF) available: ncees.orgncees.org Probability and Statistics2

3 Probability of an Event Event … a possible outcome of a trial (experiment) sun rising in the east tomorrow morning getting heads when flipping a coin baby being born as female Dr. Kinman winning Dancing with the Stars Examples:

Probability and Statistics4 Probability as a Percentage

Probability and Statistics5 Equally Likely Events We can infer the probabilities of events when all events are equally likely. experimentexample probability flipping a coin tossing a die Selecting a card from a complete deck* *but no jokers in the deck Examples:

Probability and Statistics6 Complement of an Event Venn diagram sun rising in the east tomorrow morning getting heads when flipping a coin Examples:

Probability and Statistics7 Composite event formed from 2 or more component events Component eventsComposite event Examples: Composite Event

Probability and Statistics8 Example:

Probability and Statistics9 Example:

Probability and Statistics10 Example:

Probability and Statistics11 Example:

Probability and Statistics12 Example: Roll one die and get …

from NCEES, FE Reference Handbook 13Probability and Statistics Basic Probability

Probability and Statistics14 A coin is flipped twice. What is the probability that we get heads both times?

Probability and Statistics15 A die is tossed. What is the probability that the result is an odd number?

Probability and Statistics16 A coin is flipped twice. What is the probability that there is at least one head?

Probability and Statistics17 A coin is flipped twice. What is the probability that there is at least one head?

Probability and Statistics18 A coin is flipped twice. What is the probability that either the 1 st toss is heads or the 2 nd toss is tails? The event “1 st toss heads” and the event “2 nd toss tails” are not mutually exclusive.

Probability and Statistics19 From a standard deck of cards (with no jokers), 4 cards are selected at random. What is the probability that all 4 are aces?

Probability and Statistics20 Probability and Statistics  Basic Probability  Binomial Distribution  Statistical Measures  Normal Distribution

Probability and Statistics21

from NCEES, FE Reference Handbook 22 Combinations Probability and Statistics

23 examples: special cases:

from NCEES, FE Reference Handbook 24Probability and Statistics Binomial Distribution

Probability and Statistics25

Probability and Statistics26 When Binomial Distribution is Used

Probability and Statistics27 Examples where Binomial Distribution is Used

Probability and Statistics28 A coin is flipped 4 times. What is the probability of (exactly) 3 heads?

Probability and Statistics29

Probability and Statistics30 A die is tossed 10 times. What is the probability that the die lands with the 1 face up exactly one time? A B C D.0.417

Probability and Statistics31 Ten percent of the parts in a large bin are bad. If 5 parts are selected at random, what is the probability that at least 4 of the selected parts will be good? A B C D.0.919

Probability and Statistics32 A coin is flipped 7 times. What is the probability that the number of heads is fewer than 7? A B C D.0.999

Probability and Statistics33 Probability and Statistics  Basic Probability  Binomial Distribution  Statistical Measures  Normal Distribution

Probability and Statistics34 Mean: the average of the numbers Mode: the value that occurs most often Median: the middle value The “Middle” of a Set of Measured Values Example: Measured values: 17, 9, 12, 14, 13, 18, 12, 15 Reordered values: 9, 12, 12, 13, 14, 15, 17, 18

from NCEES, FE Reference Handbook 35 Mean Probability and Statistics

from NCEES, FE Reference Handbook 36Probability and Statistics Sample Variance

Probability and Statistics37 Sample Variance for a Set of Measured Values Example: Measured values: 17, 9, 12, 14, 13, 18, 12, 15

from NCEES, FE Reference Handbook 38 Population Variance Probability and Statistics

from NCEES, FE Reference Handbook 39Probability and Statistics Standard Deviation

Probability and Statistics40 Sample Variance vs. Population Variance For both variances we calculate the difference between each value and a mean, then we square the differences and sum them, then we divide by a number. Sample Variance Population Variance

Probability and Statistics41 We have measured the following values: 17, 9, 12, 14, 13, 18, 12, 15 The mean has been modeled as What is the population variance? A.8.5 B.9.0 C.9.5 D.9.9

from NCEES, FE Reference Handbook 42Probability and Statistics Linear Regression (Least-Squares Straight Line)

Probability and Statistics43 xy Find the slope of the linear regression of the following data: A B C D.2.281

Probability and Statistics44

Probability and Statistics45 Probability and Statistics  Basic Probability  Binomial Distribution  Statistical Measures  Normal Distribution

from NCEES, FE Reference Handbook 46Probability and Statistics Normal (Gaussian) Distribution

Probability and Statistics47 Typical Problem with Normal Distribution

Probability and Statistics48 A Type I Problem: B Type II Problem: A Type III Problem: B Some Problems with Normal Distribution

Probability and Statistics49 Type I Problem: Type II Problem:

Probability and Statistics50 Type III Problem:

from NCEES, FE Reference Handbook 51Probability and Statistics Unit Normal Distribution Table

Probability and Statistics52 A set of measured values are modeled as having a normal distribution with mean 5.0 and variance 4.0. What is the probability that a new value will be larger than 5.8? A.0.15 B.0.22 C.0.28 D.0.34

Probability and Statistics53 A set of measured values are modeled as having a normal distribution with mean 7.0 and standard deviation 2.7. What is the probability that a new value will be smaller than 6.2? A.0.17 B.0.25 C.0.38 D.0.45

Probability and Statistics54 A set of measured values are modeled as having a normal distribution with mean 5.6 and variance 4.0. What is the probability that a new value will be at least 0.4 away from the mean (in either direction)? A.0.84 B.0.90 C.0.95 D.0.99

Probability and Statistics55 A set of measured values are modeled as having a normal distribution with mean 5.6 and variance 4.0. What is the probability that a new value will be within 0.4 of the mean? A.0.10 B.0.16 C.0.20 D.0.25

from NCEES, FE Reference Handbook 56Probability and Statistics Confidence Interval

from NCEES, FE Reference Handbook 57Probability and Statistics Parameter for Calculation of Confidence Interval

Probability and Statistics58 We have a set of 100 measured values for a physical quantity. We believe a normal distribution is the correct model for these data and that the variance is 9.0. However, the population mean has not yet been determined. We estimate this mean by computing the sample mean from the data, and this estimated mean is 8.3. If we want a confidence level of 95%, what is the confidence interval for the mean?