Lecture 1

Experiments, Models and Probabilities

Outline

• Definitions• Conditional Probability• Independent Trials– n choose k – multinomial coefficient

• Reliability• Matlab Experiment

Definition

• An outcome of an experiment is any possible observation of that experiment.

• The sample space of an experiment is a set of all possible outcomes.

• An event is a set of outcomes of an experiment.

• An event space is a set of events.

Telephone Usage

Question: Determine the probability of a long call.

Conditional Probability

Resistor Variability (1)

Resistor Variability(2)

Traffic Lights

Independent Trials:n choose k

Permutations

• Shuffle the deck and choose three cards in order. How many outcomes are there?– 52 x 51 x 50

Derivation

Assumptions: Four objects A, B, C, and D. Experiment: Choose two objects. How many possible outcomes? =12. Alternative Reasoning: If the order in which objects are drawn is not important (i.e. {A,B}={B,A}), then there are =6 combinations (i.e. {AB,AC,AD,BC,BD,CD}). Each combination has two permutations (i.e. {AB} is distinct from {BA}, if order matters) So 2!=Let n→4, k →2, we have Therefore,

Choose k objects out of n

Resistor Variability(2)

Transmission of Information

Transmission of Data Packets

Independent Trials:

Definitions

• For n trials of a subexperiment with sample space S={s0, ,,,sm-1}, we want to find the number observations sequences in which s0 appears n0 times, s1 appears n1 times, and so on.

Chip Fabrication

How many elements are in the entire set of possible sequences?

n0=2n1=2n2=3n3=3

Derivation

• Start with n empty slots• slots for • slots for

Multinomial Coefficient

Example

Chip Fabrication (3)

Reliability

Redundant Parts

Matlab

factorial(n)

nchoosek(n,k)

(𝑛𝑘)= 𝑛!𝑘! (𝑛−𝑘 ) !

(52)= 5 !2! (5−2 )!

=10

Flip a Fair Coin 4 Times

ceil()

Execution

Plot the histogram

Packets

Matlab Code

# of packets in simulation

n=100 n=10000 n=100000