5.1 – Random Variables
(Cover only up to Linear Functions of a Random Variable, page 225)
(Do problems 1-10 from pages 226-229)
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 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. This chapter deals with discrete random variables.
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.
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 daily temperature
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 are the same as the probabilities.
Requirements for a Probability Distribution
· 0 £ P(X = x) £ 1
· The sum of the probabilities of a discrete random variable is 1.
Mean, Variance, and Standard Deviation for a Probability Distribution
Mean
Standard Deviation
Round-Off Rules
One more decimal place than the original random variable
Note: To find the mean and standard deviation, instead of using the formulas, use the calculator by entering x into L1, P(x) into L2, and using
STAT CALC 1:1Var-Stats L1,L2
Identifying Unusual Results with the Range Rule of Thumb
Maximum usual value =
Minimum usual value =
Identifying Unusual Results with Probabilities
Unusually high: x successes among n trials is unusually high if P(x or more) is very small (such as less than 0.05)
Unusually low: x successes among n trials is unusually low if P( or fewer) is very small (such as less than 0.05)
5.2 Binomial Probabilities
Independent Events
The occurrence of one does not affect the probability of the other
Binomial Probability Distribution
A binomial probability distribution results from a procedure that meets all the following requirements:
1) The experiment has a fixed number of trials
2) The trials must be independent
3) Each trial has 2 possible outcomes
4) Probabilities remain constant for each trial
When sampling without replacement, the events can be treated as if they were independent if the sample size is no more than 5% of the population size. (That is,)
Notation and Terminology
S success
F failure
p = P(S) probability of success in one trial
q = 1 – p = P(F) probability of failure in one trial
n number of trials
r number of successes in trials
P(r) probability of EXACTLY x successes in n trials
Note: make sure that r and p both refer to the same category being called a success.
Finding Probabilities in a Binomial Experiment
1st Method: Binomial Formula
=
where r is the number of successes in n trials, p is the probability of success in any one trial, and q is the probability of failure in any one trial. (q = 1 – p)
Notice: the binomial coefficient or counts the number of branches in the tree diagram that have r successes and (n – r) failures.
Tree diagram for a binomial experiment of 3 trials
1st Trial 2nd Trial 3rd Trial Outcome
S _____ SSS
S
F ______SSF
S
S ______SFS
F
F ______SFF
S ______FSS
S
F ______FSF
F
S ______FFS
F
F ______FFF
Tree diagram for a binomial experiment of 4 trials
1st Trial 2nd Trial 3rd Trial 4th Trial Outcome
S ______SSSS
S
F ______SSSF
S
S ______SSFS
F
F ______SSFF
S
S ______SFSS
S
F ______SFSF
F
S ______SFFS
F
F ______SFFF
S ______FSSS
S
F ______FSSF
S
S ______FSFS
F
F ______FSFF
F
S ______FFSS
S
F ______FFSF
F
S ______FFFS
F
F ______FFFF
Calculator instructions for finding factorials and binomial coefficients
Evaluate factorials with the calculator:
Type number MATH PRB 4:! ENTER
Evaluate binomial coefficients with the calculator:
Type n MATH PRB 3:nCr Type r ENTER
Example: The Heart Association claims that only 10% of adults over 30 in the U.S. can pass the minimum requirements established by the Physical Fitness Commission. If 4 adults are randomly selected and given the fitness tests,
(a) What is the probability that no one passes?
(b) What is the probability that exactly one passes?
(c) What is the probability that exactly three pass?
(d) Complete a Probability Distribution Table for heart association.
(e) What is the probability that at least 2 pass?
(f) What is the probability that at most 3 pass?
Other Method: Use the Calculator
2nd VARS binompdf(n,p,x)
or 2nd VARS binompdf(n,p) to get a list of all probabilities
The option 2nd VARS binomcdf(n,p,x) gives you a cumulative probability from 0 to x.
Additional Examples:
(1) In a large office building, the probability of a defective phone is 0.05. If 20 phones are randomly selected, find the probability that:
a) Exactly 6 are defective
b) At most 3 are defective
c) At least 5 are defective
(2) 10-Questions-True/False. Probability to pass if random guesses. 60% is required to pass.
(3) 10-Questions-Sometimes/Always/Never. Probability to pass if random guess.
60% is required to pass.
(4) 10-Questions-multiple choice of 5. Probability to pass if random guesses.
60% is required to pass.
(5) What if # questions increases?
5.3 Mean, Variance, and Standard Deviation for the Binomial Distribution
In Section 5.1, we used the general formulas for any discrete probability distribution. But the special qualities of binomials distributions, lead to specialized rules for binomials
Both sets of formulas are here
Any Discrete Probability Distribution / Binomial DistributionsMean / /
Standard Deviation / /
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