Introduction to Probability

5.1 Experiments, Outcomes, Events, and Sample Spaces Sample space - The set of all possible outcomes for an experiment Roll a die Flip a coin Measure heights

Some experiments consist of a series of operations. A device called a tree diagram is useful for determining the sample space. Example: Flip a Penny, Nickel, and a Dime Event - Any subset of the sample space An event is said to occur when any outcome in the event occurs

5.2 Assigning Probabilities to Events The probability of an event A, denoted, is the expected proportion of occurrences of A if the experiment were performed a large number of times. When outcomes are equally likely Examples: Flip a fair coin Roll a balanced die

When probability is based on frequencies Example: Results of sample Males (event M) – 40 Females (event F) – 60

5.3 Some Basic Rules of Probability The closer to 1 a probability the more likely the event

5.4 Probabilities of Compound Events The complement of an event A, denoted or, is all sample points not in A. The complement rule: Joint Probability – an event that has two or more characteristics

The union of two events, denoted, is the event composed of outcomes from A or B. In other words, if A occurs, B occurs, or both A and B occur, then it is said that occurred. The intersection of two events, denoted, is the event composed of outcomes from A and B. In other words, if both A and B occur, then it is said that occurred.

5.5 Conditional Probability Sometimes we wish to know if event A occurred given that we know that event B occurred. This is known as conditional probability, denoted A|B. Example Roll a balanced green die and a balanced red die Denote outcomes by (G,R)

Red Die Green Die (1,1)(2,1)(3,1)(4,1)(5,1)(6,1) 2(1,2)(2,2)(3,2)(4,2)(5,2)(6,2) 3(1,3)(2,3)(3,3)(4,3)(5,3)(6,3) 4(1,4)(2,4)(3,4)(4,4)(5,4)(6,4) 5(1,5)(2,5)(3,5)(4,5)(5,5)(6,5) 6(1,6)(2,6)(3,6)(4,6)(5,6)(6,6)

We say the events A and B are mutually exclusive or disjoint if they cannot occur together The addition rule The conditional probability of A given B is

Example: Select an individual at random from a population of drivers classified by gender number of traffic tickets 0 tickets1 ticket2 tickets3 or more ticketsTotal Female Male Total

5.6 Independence Two events are said to be independent if the occurrence (or nonoccurrence) of one does not effect the probability of occurrence of the other. Events that are not independent are dependent.

Example: Draw two cards without replacement Multiplication rule: Suppose we return the first card thoroughly shuffle before we draw the second

Example Select an individual at random Ask place of residence & Do you favor combining city and county governments Favor (F)OpposeTotal City (C) Outside Total

5.8 Counting Techniques How many different ways are there to arrange the 6 letters in the word SUNDAY? Suppose you have a lock with a three digit code. Each digit is a number 0 through 9. How many possible codes are there?

The symbol, read as “n factorial” is defined as and so on

Evaluate each expression

Permutations Ordered arrangements of distinct objects are called permutations. (order matters) If we wish to know the number of r permutations of n distinct objects, it is denoted as In how many ways can you select a president, vice president, treasurer, and secretary from a group of 10?

Combinations Unordered selections of distinct objects are called combinations. (order does not matter) If we wish to know the number of r combinations of n distinct objects, it is denoted as In how many ways can a committee of 5 senators be selected from a group of 8 senators?