Probability Definition: randomness, chance, likelihood, proportion, percentage, odds. Probability is the mathematical ideal. Not sure what will happen in a single event but, in the long run, certain patterns emerge. We use letters like X and Y to represent quantities. These will be called random variables.

Probability Model List the outcomes for a given event (experiment or question) and associated probabilities. Example: pick a card out of a standard deck S: sample space (contains all possible outcomes) Event: single outcome or collection of outcomes S: sample space contains Event: pick a pick a

Basic Rules 1.Event A has probability P(A), which is between 0 and 1 (inclusive). 2.Probability of entire sample space, P(S), is. 3.Addition: If two events are disjoint (nothing in common), then P(A or B) =. 4.Complement: P(not A) =

Probability Model for standard deck of cards 52 cards, 4 suits (Diamonds, Hearts, Clubs, Spades) Each suit as 13 cards: 2, 3, 4, 5, …, 9, 10, J, Q, K, A P(picking any single card)= A = event that a red 5 is picked B = event that a club is picked C = event that a face card (J, Q, K) is picked P(A or B)= P(not C) =

Discrete Model If sample space is finite, the probability model is called discrete. Roll 2 six-sided dice and record the sum. List all outcomes and associated probabilities in a table. Sum Prob

Continuous Model If sample space contains a range of values, the probability model is called continuous. Density curves record probability as the area under the curve for a given range of outcomes. So, total area under the curve will always equal 1.

Continuous Model – Example 1 The uniform distribution for any real number, X, from 3 to 7 looks like:

Continuous Model – Example 2 The symmetric triangular distribution for any real number, X, from 0 to 8 looks like:

More Probability Use Venn diagrams to visualize probability rules. If events are disjoint, don’t overlap circles. Sample space, S: rectangle Events (A, B, C, …): circles inside Keep track of # of outcomes in each region of the rectangle.

Venn diagram - example Example: pick a card out of a standard deck S: sample space contains 52 outcomes (52 cards) A = event that a red 5 is picked B = event that a club is picked C = event that a face card (J, Q, K) is picked S A B C

P(A or B) S A B C S A B C P(B or C)

General Addition Rule A = event that a red card is picked B = event that a number card is picked P(A or B) S A B General Addition: P(A or B) =

Conditional Probability Rule Given a condition (you know something happened), how does that change the chances of something else happening? P(B|A)= probability of B given A S A B

Venn Diagram of 70 students C: owns a cat D: owns a dog S C D

General Multiplication Rule Rewriteto get: Experiment: pick two cards out of a standard deck

Independent Events Two (or more) events are independent if knowledge of one event does not change the chances of the other. Multiplication Rule for Independent Events:

For a cholesterol-lowering drug, there is a 5% chance that a loss-of-sleep side effect will occur. What are the chances that two people picked at random take the drug and experience sleep loss? What are the chances that at least 1out of 3 loses sleep?

The Normal Distribution Curve will capture 100% of all observations. Hence, there will be a total area of 1 below it. Then the area under the curve for a given range of values will represent the proportion (percent, fraction) of observations that fall in that range. Use curves to describe overall pattern seen in a histogram.

The proportion of scores above 80 is roughly 26.8%. The area under the density curve for scores above 80 is roughly =26.1%. Curves and proportions % v % v

Mean and Medians Location of the median on a density curve is where area under is cut in half. Location of the mean on a density curve is where the length of the curve is cut in half. On symmetric curves: On skewed curves:

Normal curves are special kinds of density curves Symmetric, single peaked, bell-shaped Use  mu  and  sigma  to talk about mean and std. dev.  –  –  -    + 

Rule About 68% of data fall within About 95% of data fall within About 99.7% of data fall within  -    +   + 2  + 3   - 2  - 3 

Example 1 Grasshopper jumps can be described by a Normal distribution with  = 12 inches and  = 2 inches. About 68% of all jumps are between inches    About where would you find the top 2.5%?

Example 1 – continued What % falls below 14 inches? What % of jumps are more than 14 inches? What % of jumps are between 14 and 16 inches?

Finding values without We use tables or calculators to find harder values, like where is the top 10% or what percent falls below a given observation. N( ,  ) means observations come from a Normal distribution with a mean of  and a standard deviation of . Standardize observation x from N( ,  ) by: The standardized value is called a

z-scores from example 1, N( ,  )

Two functions on the calculator (found under 2 nd VARS => DISTR) normalcdf( : will give area between two bounds for a given , . invNorm( : will give the observation that has a particular area to its left for a given , . normalcdf(lower bound, upper bound, ,  ) invNorm(area, ,  )  -    + 

Using the calculator with grasshopper N( ,  ) What % of jumps fall below 17 inches? normalcdf(lower bound, upper bound, ,  ) = normalcdf( ) = area below 17 = No lower bound, so: What % of jumps fall above 11.5 inches? Since total area is 1 and we have : First, find area we want normalcdf(lower bound, upper bound, ,  ) = normalcdf( ) = area =

Using the table with grasshopper N( ,  ) What % of jumps fall between 10 and inches? Calculator does this all at once with the normalcdf( function. Area between = area below – area below = - normalcdf(lower bound, upper bound, ,  ) = normalcdf( ) = area between =

Using the table with grasshopper N( ,  ) What jumps fell in the top 10%? Use invNorm function to find that observation % What observation has an area of.10 above it? What observation has an area of.90 below it? invNorm(area, ,  ) = invNorm( ) = value with.9 area below=

Using the table with grasshopper N( ,  ) Where do the middle 50% fall? Use invNorm function to find those observations % What observation has an area of below it? invNorm(area, ,  ) = invNorm( ) = value with area below = invNorm(area, ,  ) = invNorm( ) = value with area below =

Rule 1,2,3 standard deviations away accurate to two decimal places

Sampling Distributions Know the entire population: (parameter) Know only a sample (SRS): (statistic)

Law of Large Numbers - As you increase the sample size, sample mean gets closer to population mean Population = 3, 3, 8, 15, 20, 21, 22, 31, 39 Sample of size 1= 8 Sample of size 2= 8, 22 Sample of size 3= 8, 22, 31 Sample of size 4= 8, 22, 31, 3 Sample of size 5= 8, 22, 31, 3, 20

Population of 7 people and their weights (in pounds) 120 Samples of size 1: {122}, {140}, {150}, {155}, {160}, {170}, {195} Mark off the sample mean for each sample with an “x” 122, 140, 150, 155, 160, 170, x xxxxxx 120 Mark off the sample mean for each sample with an “x” x Samples of size 2: {122, 140}, {122, 150}, {122, 155}, {122, 160}, {122, 170}, {122, 195}, {140, 150}, …, (170, 195}. There are 21 possible samples. xxxxxxxxxxxx x xxxxxxx

Population of 7 people (continued) 120 Samples of size 1: 140, 122, 160, 195, 150, 155, x xxxxxx Samples of size 2: xxxxxxxxxxxxx x xxxxxxx Samples of size 6: 7 possible sample of this size. {122, 140, 160, 150, 155, 170}, … xxxxxxx

Sampling distribution of Sampling from a large population with mean  and standard deviation  : samples of size n will have their sample means distributed with a mean  and standard deviation  over root n. If population is N( ,  ), then If population is not Normal but n is large, then

Ex. 1 - Weight of eggs is N(65, 3) Your egg carton holds 9 eggs, so consider each carton as a random sample of 9 eggs. Let X be the weight of a single egg in grams and X be average weight of your carton. What is the sampling distribution for your carton’s average weight?

Weight of eggs is N(65, 3) – continued Mean weight of carton is N(65,1) Convert 67 to a z-score for a single egg: Convert 67 to a z-score for the carton:

Ex. 2 - Length of trout is N(17.5, 2.5) Your local waters contain a multitude of trout. Let X be the length of a single fish in inches and X be average length of your daily catch of five fish. What is the sampling distribution for your daily catch?

Trout length is N(17.5, 2.5) – continued Mean length of daily catch is N(17.5,1.118) Convert 16 to a z-score for a single fish: Convert 16 to a z-score for the daily catch:

Trout length is N(17.5, 2.5) – continued Mean length of daily catch is N(17.5,1.118)

Ex 3 - Length of trout is N(10, 2) Your fishing pond has another type of trout. Let X be the length of a single fish in inches taken at random and X be average length of a sample of 16 fish. What is the sampling distribution for a sample of 16 fish?

Trout length is N(10, 2) – continued Mean length of 16 fish is N(10,0.5)