Probability and Statistics

Probability and Statistics • Basic Probability • Binomial Distribution • Statistical Measures • Normal Distribution Probability and Statistics

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.org Probability and Statistics

Probability of an Event Event … a possible outcome of a trial (experiment) event Examples: Probability and Statistics

Probability as a Percentage can also be stated as The probability of is 70%. Probability and Statistics

Equally Likely Events We can infer the probabilities of events when all events are equally likely. Examples: *but no jokers in the deck Probability and Statistics

Complement of an Event is the complement of Venn diagram Examples: Probability and Statistics

Composite Event Composite event formed from 2 or more component events Examples: Probability and Statistics

and and Example: pump works, pipe intact, and *This is true as long as the and are statistically independent. Probability and Statistics

or or and Example: Jack solves problem, Jill solves problem, or and Probability and Statistics

or Alternate Solution Complement of or → and or or Example: Jack doesn’t solve problem, Jill doesn’t solve problem, or Probability and Statistics

or or or or Example: Moe has a watch, Larry has a watch, Curly has a watch, someone has a watch Probability and Statistics

or with and (and are mutually exclusive) or , when and Example: Roll one die and get … face 1 is up, face 2 is up, or Probability and Statistics

Basic Probability Probability and Statistics

A coin is flipped twice. What is the probability that we get heads both times? A. B. C. D. 1sttoss heads 2nd tossheads and and A Probability and Statistics

A die is tossed. What is the probability that the result is an odd number? A. B. C. D. face 1 is up face 3 is up face 5 is up Events , and are mutually exclusive: or or C Probability and Statistics

A coin is flipped twice. What is the probability that there is at least one head? A. B. C. D. 1sttossis heads 2ndtoss is heads WRONG: or or D (WRONG) This is wrong because events and are not mutually exclusive! Probability and Statistics

A coin is flipped twice. What is the probability that there is at least one head? A. B. C. D. The complement of at least one head is “no heads”. at least one headno heads 1sttoss tails2ndtoss tails at least one headC Probability and Statistics

A coin is flipped twice. What is the probability that either the 1sttoss is heads or the 2ndtoss is tails? A. B. C. D. The event “1sttoss heads” and the event “2ndtoss tails” are not mutually exclusive. The complement of the desired composite event is “1sttoss tails and 2ndtoss heads”, whose probability is . The desired composite event therefore has the probability . C Probability and Statistics

From a standard deck of cards (with no jokers), 4 cards are selected at random. What is the probability that all 4 are aces? A. B. C. D. 1st is ace B Probability and Statistics

Factorial Probability and Statistics

Combinations Probability and Statistics

Computing examples: special cases: Probability and Statistics

Binomial Distribution Probability and Statistics

Pascal’s Triangle: from a Diagram 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 0 1 Probability and Statistics

When Binomial Distribution is Used Binary outcomes: or with Repeated trials of with binary outcomes Underlying probabilities and do not change. Probability and Statistics

Examples where Binomial Distribution is Used 1. 2. 3. What is the probability of heads in flips of a coin? headstails A device comes off an assembly line and works with probability , and doesn’t work with probability . What is the probability that of such devices work? What is the probability that a die lands with the 1 face up times in tosses? 1not 1 Probability and Statistics

A coin is flipped 4 times. What is the probability of (exactly) 3 heads? B. C. D. heads: tails: and 3 heads 1 1 1 1 2 1 1 3 3 1 4 6 4 1 or 3 heads B Probability and Statistics

A device comes off an assembly line and works with probability , and doesn’t work with probability . What is the probability that exactly of 4 such devices works? A. B. C. D. good bad and 1good ⇒ 1good A Probability and Statistics

A die is tossed 10 times. What is the probability that the die lands with the 1 face up exactly one time? A. 0.184 B. 0.230 0.323 0.417 1 not 1 and one time one time C Probability and Statistics

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. 0.631 B. 0.720 0.853 0.919 good0.9 bad0.1 Consider and 4 good 5 good and “4 good” and “5 good” are mutually exclusive: at least 4 good4 good5 good at least 4 good 0.919 D Probability and Statistics

A coin is flipped 7 times. What is the probability that the number of heads is fewer than 7? A. 0.889 B. 0.956 0.992 0.999 heads0.5 tails0.5 fewer than 7 heads ← hard fewer than 7 heads7 heads ← easy 7 heads fewer than 7 heads 0.992 C Probability and Statistics

The “Middle” of a Set of Measured Values Mean: the average of the numbers Mode: the value that occurs most often Median: the middle value Example: Measured values: 17, 9, 12, 14, 13, 18, 12, 15 Reordered values: 9, 12, 12, 13, 14, 15, 17, 18 Mean (9 + 12 + 12 + 13 +14 + 15 + 17 + 18)/8 13.75 Mode 12 Median 13.5 Probability and Statistics

Mean Probability and Statistics

Sample Variance Probability and Statistics

Sample Variance for a Set of Measured Values Example: Measured values: 17, 9, 12, 14, 13, 18, 12, 15 8 13.75 sample variance 8.50 sample standard deviation 2.92 Probability and Statistics

Population Variance Probability and Statistics

Standard Deviation Probability and Statistics

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. Probability and Statistics

We have measured the following values: 17, 9, 12, 14, 13, 18, 12, 15 The mean has been modeled as 12.5. What is the population variance? 8.5 9.0 9.5 9.9 8 12.5 population variance 9.0 B Probability and Statistics

Linear Regression (Least-Squares Straight Line) Probability and Statistics

Find the slope of the linear regression of the following data: 1.097 1.565 1.972 2.281 y-intercept slope B Probability and Statistics

Probability and Statistics

Normal (Gaussian) Distribution Probability and Statistics

Typical Problem with Normal Distribution A physical quantity (for example, a pressure or temperature) has been measured many times. The quantity is thought to be unchanging, but the measured values are different because of noise in the measurement process. The measured values will often be modeled as having a normal distribution with mean and variance (or, equivalently, a standard deviation of ). We want to answer questions about the next (measured) value, . Probability and Statistics

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

Type I Problem: area under the curve to the right of Type II Problem: Probability and Statistics

Type III Problem: area under the curve to the right of Probability and Statistics

Probability and Statistics