Section 6.1 "Basic Concepts of Probability and Counting" Probability Experiment: An action, or trial, through which specific results are obtained Outcome: The result of a single trial in a probability experiment Sample Space: The set of all possible outcomes in a probability experiment Event: A subset of the sample space. It may consist of one or more outcomes. Identifying the Sample Space of a Probability Experiment... (1) You can use a TREE DIAGRAM. SAMPLE SPACE (12 outcomes): 1
**A simple event would be an event consisting of only one outcome, like getting heads and a 3. An example of an event that is NOT simple might be getting heads and an even number. (2) You can use the Fundamental Counting Principle. You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed below: Manufacturer: Ford, GM, Honda Car Size: compact, mid-size Color: white (W), red (R), black (B), green (G) How many different ways can you select one manufacturer, one car size, and one color? We can check with a tree diagram: 2
Some examples... (ex1) The access code for a car's security system consists of four digits. Each digit can be any number from 0 through 9. (a) How many access codes are possible when each digit can be repeated? (b) How many access codes are possible when each digit can be used only once and not repeated? (c) How many access codes are possible when each digit can be repeated, but the first digit cannot be 0 or 1? 3
(ex2) How many license plates can you make when a license plate consists of... (a) six (out of 26) alphabetical letters, each of which can be repeated? (b) six (out of 26) alphabetical letters, each of which cannot be repeated? (c) six (out of 26) alphabetical letters, each of which can be repeated, but the first letter cannot be A, B, C, or D? 4
Types of Probability... 1. Classical (or Theoretical) Probability 2. Empirical (or Experimental) Probability 3. Subjective (or Personal) Probability (1) Classical Probability is found using a ratio. It is the number of outcomes in the event divided by the total number of possible outcomes. You roll a six-sided die. Find the following probabilities: Event A = rolling a 3 P(A) = P(rolling a 3) = Event B = rolling a 7 P(B) = P(rolling a 7) = Event C = rolling a number less than 5 P(C) = P(rolling a number less than 5) = 5
(2) Empirical Probability is based on observations obtained through an experiment. It is calculated using a ratio: the number of times you observed your event (its frequency) divided by the total number of times you repeated the experiment. A company is conducting an online survey of randomly selected individuals to determine how often they recycle. So far, 2,451 people have been surveyed. What is the probability that the next person surveyed always recycles? 6
Now you try... A company is conducting a phone survey of randomly selected individuals to determine the ages of social networking site users. So far, 975 social networking site users have been surveyed. What is the probability that the next user surveyed is 23 to 35 years old? 7
As you increase the number of times a probability experiment is repeated, the empirical probability (also known as the relative frequency or percentage) of an event approaches the theoretical probability of the event. This is known as the Law of Large Numbers. A simple illustration of this involves flipping a coin and recording the probability of tossing a head: (3) Subjective Probability This probability results from intuition, educated guesses, and estimates. Based on the fact that I only studied for 20 minutes last night, the probability that I'm going to pass the test today is probably about 20%. 8
ALL probabilities - regardless of type - can only take on certain values. The probability of any event must be greater than or equal to 0 and less than or equal to 1. Symbolically: 0 P(E) 1 LASTLY... Given an event A, the complement of that event (A'... read "A prime") is the outcome of A NOT occurring. Event A = It's going to rain Complement of A = It's NOT going to rain Event B = Rolling a 2 on a die Event B' = NOT rolling a 2 on a die The sum of all probabilities must equal ONE (representing 100% of all possible outcomes), so in order to find the probability of A', you simply subtract the probability of A from 1. 9
ASSIGNMENT: Section 6.1 Worksheet (all) 10