AP Statistics Intro to Probability Sample Spaces and

AP Statistics Intro to Probability: Sample Spaces and Counting

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

Definitions n 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. n Experiment: Any sort of activity whose outcome cannot be predicted with certainty. (Flip a coin, roll a die, etc. )

Definitions n Outcome: One of the possible things that can occur as the result of an experiment. n 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). n Example: Flip three coins – what are the outcomes? n n Note: Sample spaces can be written in words and long sample spaces can be truncated.

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

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

Example n ABCDE – 5 letters taken 2 at a time. n There are 10 possible combinations: AB BC CD DE AC BD CE AD BE AE

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

Example n ABCDE – 5 letters taken two at a time in any order. n 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! n The number of permutations is not always twice the number of combinations! n Permutations can also be calculated on your calculator.

Homework n Worksheet.