AP Statistics Intro to Probability: Sample Spaces and Counting

Definitions Probability: A branch of mathematics that describes the pattern of chance outcomes. Probability: A branch of mathematics that describes the pattern of chance outcomes. Probability outcomes are the basis for inference. Probability outcomes are the basis for inference. Randomness: (not haphazardous) A kind of order that emerges in the long run when repeated events occur. Randomness: (not haphazardous) A kind of order that emerges in the long run when repeated events occur.

Definitions We call a phenomenon random if individual outcomes are uncertain, but there is none-the-less a regular distribution of outcomes in a large number of repetitions. We call a phenomenon random if individual outcomes are uncertain, but there is none-the-less a regular distribution of outcomes in a large number of repetitions. Experiment: Any sort of activity whose outcome cannot be predicted with certainty. (Flip a coin, roll a die, etc.) Experiment: Any sort of activity whose outcome cannot be predicted with certainty. (Flip a coin, roll a die, etc.)

Definitions Outcome: One of the possible things that can occur as the result of an experiment. Outcome: One of the possible things that can occur as the result of an experiment. Sample Space (S): The set of all possible outcomes. Example: Sample Space (S): The set of all possible outcomes. Example: Flip one coin: S = {H,T} Flip two coins: S = {HH, HT, TH, TT} Flip three coins: S = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT}

Tree Diagrams A Tree Diagram can help define our sample space (S). A Tree Diagram can help define our sample space (S). Example: Flip three coins – what are the outcomes? Example: Flip three coins – what are the outcomes? Note: Sample spaces can be written in words and long sample spaces can be truncated. Note: Sample spaces can be written in words and long sample spaces can be truncated.

Ways to Calculate Outcomes Fundamental Counting Principle: If there are “m” different choices for decision 1 and “n” different choices for decision 2, then the first and second decisions together can be taken “m x n” ways. Fundamental Counting Principle: If there are “m” different choices for decision 1 and “n” different choices for decision 2, then the first and second decisions together can be taken “m x n” ways.

Example On a menu, there are 5 appetizers, 10 entrees, 6 desserts, and 4 beverages. How many possible dinners are there? On a menu, there are 5 appetizers, 10 entrees, 6 desserts, and 4 beverages. How many possible dinners are there? 5 x 10 x 6 x 4 = 1200 dinners 5 x 10 x 6 x 4 = 1200 dinners

Combinations Groupings without ordering. Groupings without ordering. (order does not matter) (order does not matter) “n” objects taken “r” at a time. “n” objects taken “r” at a time. Calculator Instructions: n MATH >>> PRB 3: r ENTER Nspire: A Menu 5(Prob) 3 Comb: nCr(n,r) enter

Example ABCDE – 5 letters taken 2 at a time. ABCDE – 5 letters taken 2 at a time. There are 10 possible combinations: There are 10 possible combinations: ABBCCDDE ACBDCE ADBE AE

Permutations Grouping by order. Grouping by order. (Order matters) (Order matters) Calculator Instructions: n MATH >>> PRB 2: r ENTER Nspire: A Menu 5(Prob) 2(perm) nPr(n,r) enter

Example ABCDE – 5 letters taken two at a time in any order. ABCDE – 5 letters taken two at a time in any order. There are 20 permutations. There are 20 permutations. AB (BA)BC (CB)CD (DC)DE (ED) AC (CA)BD (DB)CE (EC) AD (DA)BE (EB) AE (EA)

Note! The number of permutations is not always twice the number of combinations! The number of permutations is not always twice the number of combinations! Permutations can also be calculated on your calculator. Permutations can also be calculated on your calculator.

Homework Worksheet. Worksheet.