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Algebra 2 Section 11.1 Notes
Permutations and Combinations
The Fundamental Counting Principle describes the method of using multiplication to count.
A permutation is an arrangement of items in a particular order. Example to choose 3 items in
order, using the fundamental counting principle, there are 3×2×1 = 6 permutations
Using factorial notation, you can write 3×2×1 as 3!, read “three factorial”. For any positive
integer n, n factorial is n! = n(n-1)(n-2)×…×3×2×1. 0! = 1
The number of permutations of n items of a set arranged r items at a time is:
n
Pr 
n!
for 0 ≤ r ≤ n
(n  r )!
A selection in which order does not matter is a combination. The number of combinations of
n items of a set arranged r items at a time is: n C r 
Algebra 2 Section 11.2 Notes
n!
for 0 ≤ r ≤ n
r!(n  r )!
Probability
Number of times the event occurs
Experimental probability of an event: P(event) =
number of trials
Sometimes you can’t run actual trials so you can estimate the experimental
probability by using a simulation. A simulation is a model of the event.
The set of all possible outcomes to an experiment or activity is a sample space.
When each outcome in a sample space has the same chance of occurring, the
outcomes are equally likely outcomes.
If a sample space has n equally likely outcomes and an event A occurs in m of these
outcomes, then the theoretical probability of event A is P( A)  m .
n
Algebra 2 Section 11.3 Notes
Probability of Multiple Events
When the occurrence of one event affects how a second event can occur, the events
are dependent events. Otherwise, the events are independent events.
For the Probability of A and B, if A and B are independent events,
then P(A and B) = P(A)×P(B).
Two events that cannot happen at the same time are mutually exclusive events. If
A and B are mutually exclusive events, then P(A and B) = 0
For the Probability of A or B, P(A or B) = P(A) + P(B) – P(A and B). If A and B
are mutually exclusive events, then P(A or B) = P(A) + P(B)
A probability distribution is a function that gives the probability of each outcome
in a sample space. You can use a table or graph to represent a probability
distribution. A probability distribution that is equal for each event in the
sample space is a uniform distribution.
Algebra 2 Section 11.4 Notes Conditional Probability
The probability that an event, B, will occur given that another event, A, has already
occurred is called a conditional probability. Conditional probability exists
when 2 events are dependent. You write the conditional probability of event
B, given that event A occurs, as P(B|A). You read P(B|A) as “the probability
of event B, given event A.”
A contingency table is a frequency table that contains data from 2 different
categories. Contingency tables and tree diagrams can help you find
conditional probabilities.
Conditional Probability: for any 2 events A and B with P(A) ≠ 0, P( B |A)  P( AandB) .
P( A)
Algebra 2 Section 11.5 Notes Probability Models
You can use probability models to analyze situations and make fair decisions. You
can use a probability model to assign probabilities to outcomes of a chance
process, which allows you to use probability and simulations to make
predictions about real-life situations.
Algebra 2 Section 11.6 Notes Analyzing Data
Statistics is the study, analysis, and interpretation of data.
Measures of central tendency, mean (average), median (# in the middle of the
ordered data) and mode (# that occurs the most often) are the most common
measures of central tendency.
A bimodal data sets has 2 modes. If a data set has more than 2 modes, then the
modes are probably not statistically useful.
An outlier is a value that is substantially different (usually much larger or much
smaller) from the rest of the data.
The range of a set of data is the difference between the greatest and least values.
If you order data from least to greatest value, the median divides the data into
2 parts. The median of each part divides the data further and you have 4 parts
in all. The values separating the 4 parts are quartiles. The interquartile
range is the difference between the third and first quartiles.
A box-and-whisker plot is a way to display data:
Minimum
Q1
Median
Q3
Maximum
A percentile is a number from 0 to 100 that you can associate with a value x from a
data set.
Algebra 2 Section 11.7 Notes
Standard Deviation
Standard deviation is a measure of how far the numbers in a data set deviate from
the mean.
A measure of variation (such as range and interquartile range) describes how the
data in a data set are spread out.
Variance and standard deviation are measures showing how much data values
deviate from the mean. The Greek letter σ (sigma) represents standard
deviation. σ2 (sigma squared) is the variance.
Variance: 
2
 (x  x)

2
n
Algebra 2 Section 11.8 Notes
(x  x)
Standard Deviation:   
2
n
Samples and Surveys
A population is all the members of a set. A sample is part of the population.
Sampling Types and Methods:
Convenience sample, select members of the population who are conveniently
and readily available.
Self-selected sample, select only members who volunteer for the sample.
Systematic sample, order the population in some way, and then select from it
at regular intervals.
Random sample, all members of the population are equally likely to be
chosen.
Study methods:
Observational study, you measure or observe members of a sample in such a
way that they are not affected by the study.
Controlled experiment, you divide the sample into 2 groups. You impose a
treatment on one group but not on the other “control” group. Then you
compare the effect on the treated group to the control group.
Survey, you ask every member of the sample a set of questions.
Algebra 2 Section 11.9 Notes
Binomial Distributions
A binomial experiment has these important features: there are a fixed number of
trials; each trial has 2 possible outcomes; the trials are independent; the
probability of each outcome is constant throughout the trials.
Algebra 2 Section 11.10 Notes Normal Distributions
A discrete probability distribution has a finite number of possible events or
values.
The events for a continuous probability distribution can be any value in an
interval of real numbers.
A normal distribution has data that vary randomly from the mean. The graph of a
normal distribution is a normal curve. A normal distribution has a symmetric
bell shape centered on the mean.
The margin of error helps you find the interval in which the mean of the
population is likely to be. The margin of error is based on the sample and the
confidence level desired. A 95% confidence level means that the probability
is 95% that the true population mean is within a range of values called a
confidence interval. It also means that when you select many different large
samples from the same population, 95% of the confidence intervals will
actually contain the population mean.
The z-score is an important measure for normally distributed data. It indicates the
number of standard deviations a value lies above or below the mean of a
population. The formula for finding the z-score of a data point of a population
is z  x   , where x is a data point, μ is the mean of the population, and σ is

the standard deviation of the population.