Math 116 - Chapter 10 - Fundamentals of Probability
Experiment: Any process that allows researchers to obtain observations
Sample Space: All possible outcomes of an experiment
Simple Event: Consists of a single outcome of an experiment
Event: Consists of one or more outcomes of an experiment
Notation for Probabilities
Probability of Event A is denoted P(A)
Round-Off-Rule for Probability: Use 3 significant digits as decimals (or use fraction form).
Finding Probabilities with the Classical Approach (Requires Equally Likely Outcomes) method
Probability Values
▪ For any event A,
▪ The probability of an impossible event is zero
▪ The probability of a certain event is one
Complementary Events
The complement of event A, denoted by (other books may use A’ or ), consists of all simple outcomes in the sample space not making up event A.
Rule of Complementary Events
Since P(A) + P() = 1
thenP(A) = 1 – P()and P() = 1 – P(A)
Law of Large Numbers: As a procedure is repeated again and again, the relative frequency probability of an event is expected to approach the actual theoretical probability.
Finding Probabilities with the Relative Frequency Approximation (experimental probability) Method
Simulation using the calculator:Simulate the experiment of rolling a die 5times by using the calculator. Then, find the experimental probability of rolling the number 6.
(Calculator instructions to generate 5 random integers from 1 to 6:
MATHPRB5:randInt(1,6,5)ENTER
Write the outcomes of your experiment here:
Use your results to find the experimental probability of rolling the number 6
A random variable is a variable (typically represented by x) that has a numeric value, determined by chance, for each possible outcome of an experiment
Examples:
The number of students passing a certain class
The average height of the students in a class
The number of girls in a family of 5 children
The sum on the faces of two rolled dice
The number of defective parts in a sample of 20
The average height of ten randomly selected female students from the class
The average daily temperature
A word about randomness
The word randomness suggests unpredictability.
Randomness and uncertainty are vague concepts that deal with variation.
A simple example of randomness involves a coin toss. The outcome of the toss is uncertain. Since the coin tossing experiment is unpredictable, the outcome is said to exhibit randomness.
Even though individual flips of a coin are unpredictable, if we flip the coin a large number of times, a pattern will emerge. Roughly half of the flips will be heads and half will be tails.
This long-run regularity of a random event is described with probability. Our discussions of randomness will be limited to phenomenon that in the short run are not exactly predictable but do exhibit long run regularity.
A discrete random variable has either a finite or a countable number of values.
A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.
A probability distribution is a graph, table, or formula that gives the probability for each possible value of the random variable.
(Notice: similar to relative frequency tables, histograms)
A probability histogram is a way to graph a probability distribution.
The vertical scale shows probabilities instead of relative frequencies.
Note that the area of these rectangles is the same as the probabilities.