Unit 2
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Lesson 1: Independent and mutually exclusive events
Terminology
A trial: operation with unpredictable results, such as coin toss
An experiment: one or more trials, such as toss a coin for twice
An outcome: result of an experiment, such as one tail and one head of coin tosses
The sample space: the set of all possible outcomes, {HH, HT, TH, TT}
An event: one or more outcomes, such as {HT, TH} (one head}, {HH, TH} (the last toss is a head), {HH, HT, TH} (at least one head) …
Probability: likelyhood, between 0 to 1, p(event)=|event|/|sample space|
Impossible event: p=0
Certain event: p=1
Complementary event: A and A', p(A)+p(A^′ )=1
Addition Rule for Probability
Independence and conditional probability
Independent events
P(A∪B)=P(A)+P(B)−P(A∩B)
Mutually exclusive events
Lesson 2: Counting Principles
Fundamental Counting Principle
Rule of product
Rule of sum
Permutations
〖(_n^)P〗_r=n×(n−1)×…×(n−r+1)=n!/(n−r)!
Divide out the duplicates
n!/a!b!c!
Combinations
〖(_n^)C〗_r=n!/r!(n−r)!
Lesson 3: Tree diagrams and combined events
Lesson 4: Discrete probability distributions
Random variable: number form of possible outcomes
x 0 1 2
Event for x
P(X=x)
Lesson 5: Binominal Probability Distributions
x 0 1 2 … i … n
P(X=x) 〖(_n^)C〗_0 p^0 q^n 〖(_n^)C〗_1 p^1 q^(n−1) 〖(_n^)C〗_2 p^2 q^(n−2) 〖(_n^)C〗_i p^i q^(n−i) 〖(_n^)C〗_n p^n q^0
Lesson 5n: Normal Probability Distributions
Continuous probability distribution
Normal random variable and normal distribution
Standard deviation σ: measures the spread of the distribution
Spread of different values.
Bigger standard deviation -> more spread -> flatten curve
Properties of Normal Distribution: p(X<x)=f(x, μ, σ)
Curve is symmetrical about the mean μ
Mean divide area into halves
Total area under curve is 1
μ and σ defines the shape of the curve
Standard normal distribution, μ=0, σ=1
Converting any normal distribution of X into standard normal distribution Z
z=(x−μ)/σ
Lesson 6: Expected value
x x_1 x_2 x_3
Event for x
P(X=x) P(x_1) P(x_2) P(x_3)
Value for x v_1 v_2 v_3
Expected value: E(X)=v_1 P(x_1 )+v_2 P(x_2 )+v_3 P(x_3 )=Σv_i P(x_i)