Probability and Statistics

Probability Experimental Probability Sample Space P(event)=number of times the event occurs number of trials Sample Space Sample Space Event

Events Independent Dependent Mutually Exclusive When the outcome of one event does NOT effect the outcome of a second event Dependent When the outcome of one event affects the outcome of a second event Mutually Exclusive When two events cannot happen at the same time

Probability Probability of A and B If A and B are independent P(A and B)= P(A)*P(B) Probability of A or B If mutually exclusive P(A or B)= P(A) + P(B) If NOT mutually exclusive P(A or B)= P(A) + P(B)- P(A and B)

Probability Distributions Frequency table is a list of the outcomes in a sample space and the number of times each occurs Probability Distribution is a function that gives the probability of each event in a sample space

Conditional Probability Contains a condition that may limit the sample space fro an event Notation: P(B|A) The probability of event B given event A P(B|A)=

Homework Problems Practice 1-6 Practice 9-7 Practice 12-1 # 2, 3 Practice 9-7 # 1-8, 11, 14, 17 Practice 12-1 # 1, 2 Practice 12-2 # 2, 5

Probability and Analyzing Data

Counting Principle Used when a number of objects are being chosen Multiply all possible choices together Ex: Sally has 4 shirts, 12 pants, and 13 pairs of choose to choose from. How many outfits can be created? 4*12*13

Permutations and Combinations Order matters with Permutations Order does not matter with combinations Permutations nPr Combinations nCr

Examples Seven yachts enter a race. First, second, and third places will be given to the three fastest yachts. How many arrangements of first, second, and third places are possible with seven yachts? A reading list for a course in literature has 20 books on it. In how many ways can you choose four books to read?

Homework Reteaching 6-7 Practice 6-7 # 4-7 # 1-4, 10, 11, 12, 13, 16, 25, 27

Analyzing Data

Measures of Center Mean Median Mode Average of all values in the data set Median Middle value when data set is in numerical order If the data set has an even number of values then average the two middle terms Mode Most frequent value in the data set

Measures of center Put your data set in L1 Use 1-Var Statistics under stat, calc

Measures of Spread Range Standard Deviation Inter Quartile Range (IQR) Maximum value- minimum value Standard Deviation σx in 1-var stats Inter Quartile Range (IQR) Q3-Q1

Quartiles Divides the data set into four equal parts Q1- median of lower half of the data set Q2- median of the data set Q3- median of upper half of the data set

Displaying Data Box and Whisker Plots Uses minimum, Q1, Q2, Q3, and maximum to create a box and whisker plot These can also be graphed in the calculator under stat plot

Percentiles Divides the data set into two parts One part above the percentile and one part below Ex: If a number is at the 63rd percentile then about 63% of the data is below this number

Using Percentiles Example 54, 98, 45, 87, 98, 64, 21, 61, 71, 82, 93, 65, 62, 98, 87, 24, 65, 91, 31, 47 Order the data set 1st! 21, 24, 31, 45, 47, 54, 61, 62, 64, 65, 65, 71, 82, 87, 87, 93, 97, 98, 98, 98 Find the values of the 20th and 65th percentile (n=20) 20*.20=4 20*.65=13 47 is the 20th percentile 87 is the 65th percentile

Finding Outliers Outliers are values in the data set that are substantially different than the rest of the data set To find outliers: 1. Calculate IQR 2. Multiply IQR times 1.5 3. Subtract the value from step 2 from Q1 4. Add the value from step 2 to Q3 Any values in the data set below or above step 3 and 4 are outliers!

Example 56, 65, 73, 59, 98, 65, 59 IQR= 73-59 = 14 1.5(14) = 21 59-21 = 38 73+21 = 94 Any number below 38 or above 94 is an outlier….so 98 is an outlier

Homework Practice 12-3 Practice 12-4 # 1, 3, 5, 7, 22, 27, 29 # 1, 10, 15 parts a-d

Working with Samples

Recall Sample is a portion of a population Samples can be bias based on who and where people are asked A sample can be used to find a sample proportion A ratio of the number of times an event occurs in a sample size of n

Choosing a sample If every member of the population is equally likely to be chosen then it is a random sample The larger the sample the smaller margin of error A margin of error is calculated by

Homework Practice 12-5 # 2 - 7, 11, 13

Binominal Distribution

Binominal Experiment Situation involves repeated trials Each trial has two possible outcomes Success or failure The probability of success is constant throughout the trials The trials are independent Tree diagrams are used to analyze binominal probabilities

Binominal Probability If there are repeated independent trials each with a probability of success, p, and a probability of failure, q. Then the probability of x successes in n trials is…

Example A calculator contains four batteries. With normal use each batteries has a 90% chance of least 1 year. What is the probability that all four batteries will last a year? n=4 x=4 p=.9 q=.1

Homework Practice 12-6 # 3, #5 -8

Normal Distributions

Normal Distributions Shows data that vary randomly from the mean The pattern of data is bell shaped This is called a normal curve Standard Normal Curve A normal distribution centered on the y-axis Mean = 0 Standard Deviation = 1

68% -95%- 99.7% When the data set is normally distributed… 68% of the data falls within 1 standard deviation from the mean 95% of the data falls within 2 standard deviations from the mean 99.7% of the data falls within 3 standard deviations from the mean

Homework Practice 12-7 # 5, 6 ,11, 12, 13, 16