Math 130 Review Sheet
6 Introductory Counting and Probability
Notation
c∈A
(c is an element of set A)
c∉A
(c is not an element of set A)
A⊂B
(A is a subset of B)
M∩E
(M intersect E)
M∪E
(M union E)
𝑀∁
(M complement)
𝑈
(Universe)
∅
(The “null set” or “empty set”)
𝓃(𝐴)
(The count of A)
𝑃[𝐴]
(Probability of A)
𝑆
(Sample space)
Formulas and Definitions
Event: Subset of the sample space.
Additive Law of Probability: 𝑃[𝐴 ∪ 𝐵] = 𝑃[𝐴] + 𝑃[𝐵] − 𝑃[𝐴 ∩ 𝐵]
Complement Rule: 𝑃[𝐴] = 1 − 𝑃[𝐴∁ ]
The Fundamental Counting Principle (FCP): If a first task has 𝑎 possible completions, a second has 𝑏 possible
completions, and a final task has 𝑧 possible completions, then the number of possible combined completions for
all tasks is 𝑎 ∙ 𝑏 ∙…∙ 𝑧.
Relative Frequency:
# of times event E occurs
# of times experiment is repeated
7 More Counting and Probability
Notation
P [A|B]
(Probability of A given B)
Formulas and Definitions
The Trivial Subsets of S: If S is any set, then S is a subset of S (S⊂S), and ∅ is a subset of S (∅ ⊂ S).
Proper Subsets of S: All other subsets of S.
Permutations (of a set of size 𝒏): The number of possible orderings (permutations) of a set of any size 𝑛 is 𝑛! =
𝑛(𝑛 − 1)(𝑛 − 2) ∙ … 3 ∙ 2 ⋅ 1. (Note:0! = 1)
Permutations (of 𝒏 things taken 𝒓 at a time): The number of possible permutations of 𝑟 objects chosen from 𝑛
objects is
𝑛!
(𝑛−𝑟)!
abbreviated 𝑃(𝑛, 𝑟) or 𝑛𝑃𝑟.
Conditional Probability: P[A|B] =
P[A∩B]
P[B]
; P[B] ≠ 0
Combinations (of 𝒏 things taken 𝒓 at a time): The number of possible subsets of size 𝑟 which could be chosen
𝑛!
from a set of size 𝑛 is (𝑛−𝑟)!𝑟! abbreviated 𝐶(𝑛, 𝑟) or 𝑛𝐶𝑟.
Permutations of “Some Alike” Objects: The number of permutations of a set of 𝑛 objects, with 𝑛1 of type #1, 𝑛2
of type #2, …, 𝑛𝑘 of type #𝑘, is
𝑛!
𝑛1 !𝑛2 !…𝑛𝑘!
Independent Events: Events A and B are independent if the following is true: 1. P[A|B]= P[A] or 2. P[B|A]= P[B].
Events that are not independent are dependent.
The Multiplicative Rule: P[A ∩ B] = P[A] ∙ P[B|A]
Probability for Two Independent Events: P[A ∩ B] = P[A] ∙ P[B]
Mutually Exclusive Events: Mutually Exclusive Events are events whose intersection is empty (𝐴 ∩ 𝐵 = ∅).
Reliability of a Series System: If a series system contains 𝐾 independent components with reliabilities given by
𝑝1, 𝑝2, … , 𝑝𝑘 .Then the reliability of the series system, denoted by 𝑅𝑠 , is the product of the individual reliabilities:
𝑅𝑠 = 𝑝1∙ 𝑝2∙ … ∙ 𝑝𝑘 .
Reliability of an 𝒏-compenent Parallel System: If a parallel system consists of 𝑛 independent components, each
having reliability 𝑝, then the parallel system’s reliability is given by: 𝑅𝑠 = 1 − (1 − 𝑝)𝑛 .
Problem Solving
Solving Probability Problems in Tree Diagram Form:
1.
2.
3.
Represent sample space with tree diagram. Label each of the paths on the tree. Attach a probability to each branch.
Find the probability for each path by multiplying all the branch probabilities that lie on that path.
The probability of the event of interest is found by adding the probabilities of its paths.
8 Random Variables and Probability Distributions
Notation
E(x)
(Expected value of x)
Σ
(Sum of)
𝜇𝑥
(Mean of x)
𝜎𝑥
(Standard Deviation of x)
𝜎𝑥2
(Variance of x)
Formulas and Definitions
Mean of a Probability Distribution: When a population of values of a random variable 𝑋 is given in summary form
by a probability distribution, then the mean of 𝑋 (or the mean of the probability of distribution of 𝑋) is given by:
𝜇𝑥 = Σx ∙ 𝑝(𝑥).
Expected Value: If 𝑋 is a random variable whose probability at 𝑥 is given by 𝑝(𝑥), then the expected value 𝑓(𝑥) is
given by: 𝐸[𝑓(𝑋)] = Σ𝑓(𝑥) ∙ 𝑝(𝑥).
Population Variance: The population variance 𝜎𝑥2 may be computed by finding 𝐸[𝑥 2 ] − 𝜎𝑥2 . That is:
𝜎 2 = Σ𝑥 2 ∙ 𝑝(𝑥) − 𝜇𝑥2
Standard Deviation: 𝜎 = √𝜎 2
Chebyshev’s Theorem: If 𝑘 is any real number greater than 1, than at least 1 −
1
𝑘2
of the values in the population
are with in 𝑘 standard deviations of the mean. Equivalently, 𝑃[𝜇𝑥 − 𝑘 ∙ 𝜎𝑥 < 𝑋 < 𝜇𝑥 + 𝑘 ∙ 𝜎𝑥 ] ≥ 1 −
1
𝑘2
Binomial Probability Distribution: In a binomial experiment, the probability of 𝑟 successes in 𝑛 independent trials,
where 𝑝 is the success probability on each trial, and 𝑟 can be any of the possible values 0, 1, …,𝑛, is given by:
𝑝(𝑟) = 𝐶(𝑛, 𝑟) ∙ 𝑝𝑟 ∙ (1 − 𝑝)𝑛−𝑟
Binomial Mean and Standard Deviation: The mean and standard deviation for a binomial distribution is: 𝜇 = 𝑛 ∙ 𝑝
and 𝜎 = √𝑛 ∙ 𝑝 ∙ (1 − 𝑝).
Normal Distribution:
(Note: 34.13%=0.3413)
Z-Score: The z-score (or standard score) of a value of 𝑥 measures how far 𝑥 is from its mean in terms of standard
deviations. Specifically: 𝑧 =
𝑥−𝜇
𝜎
9 Introduction to Statistics
Notation
𝑋̅
(Sample mean)
𝑆2
(Sample variance)
𝑀
(Sample median)
𝑆
(Sample standard deviation)
Formulas and Definitions
Sample Mean: If 𝑥1 , 𝑥2 , … , 𝑥𝑛 represent the 𝑛 items in a random sample of size 𝑛, then the sample mean (𝑋̅), is
their sum divided by 𝑛: 𝑋̅ =
Σxi
𝑛
Sample Median: The sample median (𝑀), is the middle number, or the average of the two middle numbers.
Sample Mode: The sample mode is the most frequent value in the sample.
Sample Range: The sample range is the largest value minus the smallest value.
2
Sample Variance: 𝑆 =
Σ𝑥 2 −
(Σ𝑥2 )
𝑛
𝑛−1
Sample Standard Deviation: 𝑆 = √𝑆 2
Approximation for 𝑺:
Sample Size 𝑛
Range Approximation for 𝑆
𝑛 ≥ 200
50 ≤ 𝑛 ≤ 199
16 ≤ 𝑛 ≤ 49
𝑛 ≤ 15
𝑅𝑎𝑛𝑔𝑒
6
𝑅𝑎𝑛𝑔𝑒
5
𝑅𝑎𝑛𝑔𝑒
4
𝑅𝑎𝑛𝑔𝑒
√𝑛
̅ : Consider a value whose mean is 𝐸(𝑋) = 𝜇𝑥 and whose
Mean and Variance for Sampling Distribution of 𝑿
2
variance is 𝜎 . The mean and variance of the sampling distribution 𝑋̅ is computed from a random sample of size 𝑛
̅ : 𝐸(𝑋̅) = 𝐸(𝑋) = 𝜇; Variance of 𝑿
̅ : 𝜎𝑥̅2 =
taken with replacement given by: Mean of 𝑿
Standard Error of the Mean: 𝜎𝑥̅ =
𝜎
√𝑛
𝜎2
𝑛