|PART THREE Essentials| -- Permutations and Combinations
Permutation--a set of items in which the order is important
Without replacement--duplicate items are not permitted
With replacement--duplicate items are permitted
Permutations, both with and without replacement, can be computed by using the "sequential" method instead of the formula. This provides way of verifying the formula result.
Combination--a set of items in which the order is not important
Without replacement--duplicate items are not permitted
With replacement--duplicate items are permitted
In the formulas, "n" designates the number of items available, from which "r" is the number that will be chosen. (Can r ever exceed n?)
To apply the correct formula when confronting a problem, two decisions must be made:
Is order important (permutations) or not (combinations)?
Are duplicates permitted (with replacement) or not (without replacement)?
Lotteries
Usually combination ("Lotto") or permutation ("Pick 3” or “Pick 4") cases
Lotto games are usually without replacement--duplicate numbers are not possible
Pick 3 or 4 games are usually with replacement--duplicate numbers are possible
Poker hands
Can be computed using combinations and the relative frequency method
Can also be computed sequentially
|Essentials| -- Mathematical Expectation
Discrete variable--one that can assume only certain values (often the whole numbers)--there is only a finite
countable number of values between any two specified values
Examples: the number of people in a room, your score on a quiz in this course, shoe sizes (certain fractions permitted), hat sizes (certain fractions permitted)
Continuous variable--one that can take on any value--there is an infinite number of values between any two specified values
Examples: your weight (can be any value, and changes as you breathe), the length of an object, the amount of time that passes between two events, the amount of water in a container (but if you look at the water closely enough, you find that it is made up of very tiny chunks--molecules--so this last example is really discrete at the submicroscopic level, but in ordinary everyday terms we would call it continuous)
Mean (expected value) of a discrete probability distribution
Probability distribution--a set of outcomes and their likelihoods
Probability-weighted average of the outcomes
Each outcome is multiplied by its probability, and these are added.
The result is not an estimate. It is the actual population value, because the probability distribution specifies an entire population of outcomes. ("μ" may be used, without the estimation caret above it.)
The mean need not be a possible outcome, and for this reason the term "expected value" is misleading.
Variance of a discrete probability distribution
Probability-weighted average of the squared deviations (similar to MSD)
Each squared deviation is multiplied by its probability, and these are added.
The result is not an estimate. It is the actual population value, because the probability distribution specifies an entire population of outcomes. ("σ2" may be used, without the estimation caret above it.)
Standard deviation of a discrete probability distribution--the square root of the variance
("σ" may be used, without the estimation caret ^ above it.)
|Essentials| -- The Binomial Distribution
Binomial experiment requirements
Two possible outcomes on each trial
The two outcomes are (often inappropriately) referred to as "success" and "failure."
n identical trials
Independence from trial to trial--the outcome of one trial does not affect the outcome of any other trial
Constant p and q from trial to trial
p is the probability of the "success" event
q is the probability of the "failure" event; (q = (1-p) )
"x" is the number of "successes" out of the n trials.
Symmetry is present when p = q
When p < .5, the distribution is positively skewed (high outliers).
When p > .5, the distribution is negatively skewed (low outliers).
Binomial formula--for noncumulative probabilities
Cumulative binomial probabilities--computed by adding the noncumulative probabilities
Binomial probability tables--may show cumulative or noncumulative probabilities
If cumulative, compute noncumulative probabilities by subtraction
Parameters of the binomial distribution--n and p
Binomial formula: P(x) = n!/(x!(n-x)! * p^x * q^(n-x)
Note that when x=n, the formula reduces to p^n, and when x=0, the formula reduces to q^n.
These are just applications of the multiplicative rule for independent events.
Terminology--explain each of the following:
PERMUTATIONS AND COMBINATIONS: permutations, permutations with replacement, sequential method, combinations, combinations with replacement. MATHEMATICAL EXPECTATION: random variable, discrete variable, continuous variable, probability distribution, probability histogram, mean of a probability distribution, variance and standard deviation of a probability distribution, probability-weighted average of outcomes (mean), probability-weighted average of squared deviations (variance). BINOMIAL DISTRIBUTION: binomial experiment, requirements for a binomial experiment, independent trials, binomial probabilities, cumulative binomial probabilities, binomial distribution symmetry conditions, binomial distribution skewness conditions, binomial distribution parameters, mean and variance of a binomial distribution
Skills and Procedures--given appropriate data,
PERMUTATIONS AND COMBINATIONS:
decide when order is and is not important
decide when selection is done with replacement and without replacement
compute permutations with and without replacement using the permutation formula
compute combinations with and without replacement using the combination formula
use the sequential method to compute permutations with and without replacement
solve various applications problems involving permutations and combinations
MATHEMATICAL EXPECTATION:
compute the mean, variance, and standard deviation of a discrete random variable
solve various applications problems involving discrete probability distributions
BINOMIAL DISTRIBUTION:
compute binomial probabilities and verify results with table in textbook
compute cumulative binomial probabilities
compute binomial probabilities with p = q and verify symmetry
solve various application problems using the binomial distribution
Concepts
PERMUTATIONS AND COMBINATIONS:
give an example of a set of items that is a permutation
give an example of a set of items that is a combination
tell if, in combinations/permutations, "r" can ever exceed "n"
MATHEMATICAL EXPECTATION
give an example (other than water) of something that looks continuous at a distance, but, when you get up close, turns out to be discrete
explain why "expected value" may be a misleading name for the mean of a probability distribution
describe how to compute a weighted average
BINOMIAL DISTRIBUTION:
explain why rolling a die is or is not a binomial experiment
explain why drawing red/black cards from a deck of 52 without replacement is or is not a binomial experiment
explain why drawing red/black cards from a deck of 52 with replacement is or is not a binomial experiment