Section 6.2: Probability Models Ways to show a sample space of outcomes of multiple actions/tasks: (example: flipping a coin and rolling a 6 sided die) 1)Tree diagram Task #1 ↑ Task #2 ↑ 2)Multiplication (or Counting) Principle If the 1 st task can happen n ways and the 2 nd can happen m ways, then are nm outcomes of the task together. So there 2 results on a coin and 6 results for a die, so there are 6 2 = 12 possible outcomes together.

3)Listing systematically (organized list) (for the example of flipping a coin and rolling a die) List all the possible outcome that result in heads followed by all the outcomes that result in tails OR all the outcomes that have a roll of 1, followed by 2, and so on. Statistics (specifically probability) is not all about flipping coins and rolling dice, but the outcomes can simulate many real world random phenomenon.

If selecting objects from a collection of distinct choices (ex: drawing playing cards), there are two important situations that must be considered: Selecting WITH replacement: Choose, record, put back Selecting WITHOUT replacement: Choose, record, keep How many ways can a 3 digit numbers be made? Using the counting principle: ____________ = 1 st 2 nd 3 rd digit digit digit ways Using the counting principle: ____________ = 1 st 2 nd 3 rd digit digit digit ways

A permutation of n objects using only r of those objects in each arrangement is written as n P r. This is if you only use SOME of the objects. In a race with eight horses, how many ways can 1 st, 2 nd, and 3 rd place be awarded? 8P38P3 = 336 This can also be done using the counting principle… 8 * 7 * 6 = st 2 nd 3 rd

A combination is an arrangement where the order does not matter (ABC is the same as BCA). A combination of n object taken r at a time is symbolized as n C r. Example: A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire 4 of the bidding companies. How many different combinations of the 4 companies can be selected? 16 C 4 = 1820

Rules of Probability 1) 0 ≤ P(E) ≤ 1 2)P(S) = 1 {S represents the entire sample space of outcomes} 3)If events A & B are disjoint (cannot BOTH happen in one trial) P(A or B) = P(A) + P(B) 4)P(not E) = 1 – P(E) 5)If events A & B are independent, P(A and B) = P(A) P(B) Set Notation for Probability: = A or B = A and B = not E

When events are not disjoint (can happen simultaneously), the probability of their union (A or B) is calculated as follows: P(A or B) = P(A) + P(B) – P(A and B) or P(A U B) = P(A) + P(B) – P(A ∩ B) Example: 55% of adults drink coffee 25% of adults drink tea 45% of adults drink cola 15% drink both coffee and tea 5% drink all three 25% drink coffee and cola 5% drink only tea What percent drink cola and tea? What percent drink NONE of these? Cf T Cl