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ENGR 224/STAT 224 Probability and Statistics Lecture 5. Probability Theory. Definition: An experiment is any process that allows researchers to obtain observations. Definition: An event is any collection of results or outcomes of an experiment.
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Probability Theory • Definition: An experiment is any process that allows researchers to obtain observations. • Definition: An event is any collection of results or outcomes of an experiment. • Definition: A simple eventis an outcome or an event that cannot be broken down any further.
Example: Rolling a Die • Rolling a die is an experiment. • An event is rolling a 3. • Rolling a 3 is a simple event, since it cannot be broken further.
Example: Rolling a pair of dice • Rolling two dice is an experiment. • An event is rolling a 7. • Rolling a 7 is not a simple event, since it can be broken into the simple events of rolling a (6,1) (5,2) (4,3) …… etc.
Definition:Sample Space The sample space for an experiment consists of all simple events. That is it consists of all outcomes that cannot be broken down further. Example: Rolling a single die. The sample space is made of 6 simple events: 1, 2, 3, 4, 5, 6
Example: Rolling a pair of dice. The sample space is made of 36 simple events: 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6
Example: Flashlight Batteries • Ex 2.4 – If a new type D flashlight battery has a voltage that is outside certain limits, that battery is characterized as a failure (F); if the battery has a voltage within the prescribed limits, it is a success (S). Suppose an experiment consists of testing each battery as it comes off an assembly line until we first observe a success. Describe the Sample Space, and the Event of having a success in the first three batteries. S = { S, FS, FFS, FFFS, … } A = {S, FS, FFS}
Definition:Probability (Experimental) The probability of an event A, P(A), is defined to be
Definition:Probability (Theoretical) The probability of an event A, P(A), is defined to be the sum of probabilities assigned to simple events that make up A. If A is the set of all successes, and S is the sample space then
Example: Playing Cards Consider a standard deck of playing cards. What is the probability of obtaining a spade. • Theoretically? • Experimentally? P(spade)=0.25
Law of Large Numbers As an experiment is repeated over and over again, the experimental probability approaches the theoretical probability.
In Class Exercises: • Problem 3 page 50 • Problem 5 page 51
Example: Rolling a Die Consider the Questions. • What is the probability of obtaining a 6 or an even number with 1 roll of a die? • What is the probability of obtaining a 6 or an odd number with 1 roll of a die? Write out the sample space and the event space for each, and calculate the probability P(6 or even) = 1/2 P(6 or odd) = 2/3
B A Definition: P(A or B) = If A and B are two events, then P(A or B) = P(A)+P(B) - P(A and B) A and B
Definition: Mutually Exclusive Two events are mutually exclusive if P(A and B) = 0. Examples: • Rolling a 6 or an odd number are mutually exclusive events with a single roll of a die. • Choosing a heart or a club are mutually exclusive events. • Choosing a heart or an Ace are NOT mutually exclusive.
Example: Rolling a die P(6 or even number)= P(even)+P(6)-P(even and six) but we note that P(even) =3/6 P(6) = 1/6 and P(6 and even)=1/6 Therefore P(6 or Even) = 1/6+3/6-1/6 = 1/2
Example: Playing Cards What is the probabilitydrawing a heart or a queen in one flip of a card. Solution: P(heart or queen)=P(heart) +P(queen) - P(queen and heart) P(heart or queen)=13/52 + 4/52 - 1/52 P(heart or queen)=16/52
Law of Complements If A is an event with probability, P(A), then the probability of an event not A is P(not A) = 1 - P(A)
Example: Undergraduate Students at StFX • It is known that there are 2000 students (1000 of whom are female) in the BBA progam, an additional 2000 students (1200 of whom are female) in the Arts program, and finally1000 students in Science (400 of whom are female). • What is the probability that a student chosen at random is in the Arts program? • What is the probability that a student chosen at random is Female? • What is the probability that a student chosen at random is Female or in the Arts Program? • What is the probability that a student chosen at random is Male or Female? • What is the probabilty that a student chosen at random is not in the Science program? • What is the probabilty that a student chosen at random is not in Science or a Female?
Two Way Tables • One way to complete the previous problem is to complete a “TWO-WAY” table where one of the variables is represented by rows, and the other in columns. • One can also represent the same data using percentages.
Descriptive Phrases Descriptive Phrases require special care! • At most • At least • No more than • No less than
Practice Problems • Problem 15 page 58 • Problem 21 page 58
Overview • Definitions for • Probability, Experiment, Event, Sample Space, Simple Event • Law of large Numbers • If A and B are two events then P(A or B) = P(A) + P(B) - P(A and B) • Two events A and B are mutually exclusive if P(A and B) = 0. • If A is an event with probability P(A), then P(not A) = 1 - P(A).
Homework Reread Chapter 2
