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Unit 3: Probability
 You
will need to be able to describe how you
will perform a simulation




Create a correspondence between random
numbers and outcomes
Explain how you will generate your random
numbers and when you will know to stop
Make sure you understand the purpose of the
simulation
With or without replacement?
 Probability
is a
measurement of the
likelihood of an event. It
represents the
proportion of times we’d
expect to see an
outcome in a long series
of repetitions.
 P(event)
# 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠
=
# 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒
 Addition
Rule:
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
 Multiplication Rule:
𝑃 𝐴 ∩ 𝐵 = 𝑃(𝐴) × 𝑃(𝐵|𝐴)
 Conditional Probability:
𝑃 𝐴𝐵 =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
 Disjoint
(mutually exclusive) basically means
no outcomes in common.
 Two events are independent if the
occurrence of one has no effect on the
probability that the other event occurs.
 Two tests for independence:


𝑃 𝐴 𝐵 = 𝑃 𝐴 or
𝑃 𝐴 𝑎𝑛𝑑𝐵 = 𝑃(𝐴) × 𝑃(𝐵)
 When
calculating probabilities, it helps to
consider the Sample Space.



List all outcomes if possible.
Draw a tree diagram or Venn diagram
Use the Multiplication Counting Principle
 Sometimes
it is easier to use common sense
rather than memorizing formulas!
This chapter introduced us
to the concept of a random
variable. We learned how
to describe an expected
value and variability of
both discrete and
continuous random
variables.
A
Random Variable, X, is a variable whose
outcome is unpredictable in the short-term,
but shows a predictable pattern in the long
run.
 Discrete vs. Continuous
 The
Expected Value, E(X)=μ, is the long-term
average value of a Random Variable.
 E(X) for a Discrete X
 E(X) = μ =
𝑥 ∙ 𝑝(𝑥)
X
1
5
20
P(X)
0.5
0.2
0.3
μ = 1(0.5) + 5(0.2) + 20(0.3)
= .5 + 1+ 6
= 7.5
 The
Variance, 𝑉𝑎𝑟(𝑋) = 𝜎 2 , is the amount of
variability from μ that we expect to see in X.
 The
Standard Deviation of X, 𝜎 =
X
1
5
20
P(X)
0.5
0.2
0.3
𝑉𝑎𝑟 𝑋 = 𝜎 2 =
𝑥−𝜇
𝑉𝑎𝑟 𝑋 = 69.25
𝜎 = 8.32
2
∙ 𝑝(𝑥)
𝑉𝑎𝑟(𝑋)
 The
following rules are helpful when working
with Random Variables.
𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑥
𝜇𝑋±𝑌 = 𝜇𝑋 ± 𝜇𝑌
2
𝜎𝑎+𝑏𝑋
= 𝑏2 𝜎𝑥2
2
𝜎𝑋±𝑌
= 𝜎𝑋2 + 𝜎𝑌2
 Some
Random Variables are the result of
events that have only two outcomes (success
and failure).
 We define a Binomial Setting to have the
following features




Two Outcomes - success/failure
Fixed number of trials - n
Independent trials
Equal P(success) for each trial
 If
X is B(n,p), the following formulas can be
used to calculate the probabilities of events
in X.
P(X = k) = binompdf (n, p, k)
P(X ≤ k) = binomcdf (n, p, k)
 If
conditions are met, a binomial situation
may be approximated by a normal
distribution
 If np>10 and n(1-p)>10, then B(n,p) ~ Normal
 Some
Random Variables are the result of
events that have only two outcomes (success
and failure), but have no fixed number of
trials.
 We define a Geometric Setting to have the
following features:




Two Outcomes - success/failure
No Fixed number of trials
Independent trials
Equal P(success) for each trial
 If
X is Geometric, the following formulas can
be used to calculate the probabilities of
events in X.
𝑃 𝑋 =𝑘 = 1−𝑝
1
𝜇𝑥 =
𝑝
𝜎𝑥 =
𝑘−1 𝑝
= 𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓(𝑝, 𝑘)
1−𝑝
𝑝2
𝑃 𝑋 >𝑘 = 1−𝑝
𝑘
A
sampling distribution is the distribution of
all samples of size n taken from the
population.
 Be able to describe center, shape and spread
 When
we take a sample,
we are not guaranteed the
statistic we measure is
equal to the parameter in
question.
 Further, repeated sampling
may result in different
statistic values.
 Bias and Variability
The center of the sampling distribution is equal
to the population mean
 The shape of the sampling distribution is
approximately Normal:




If the population is approximately Normal OR
The sampling distribution becomes more normal as
the sample size increases (CLT)
The standard deviation of the sampling
distribution is called the Standard Error (SE) and
𝜎
can be found by 𝜎𝑋 =
𝑛

Can use Normal techniques to solve problems
𝑥−𝑚𝑒𝑎𝑛
𝑥−𝜇
with 𝑧 =
= 𝜎
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑛
 The
center is equal to p, the population
proportion.
 The
standard deviation is 𝜎𝑝 =
𝑝(1−𝑝)
𝑛
 As
the sample size gets larger, the shape of
the sampling distribution gets more normal.


np > 10 and
n(1-p) > 10
 Can
use normal distribution as an
𝑝−𝑝
approximation with 𝑧 =
𝑝(1−𝑝)
𝑛