1. Probability Theory Basic concepts Probability Permutations
Combinations

2. Randomness Experiments Events Trial: execution of one experiment
Outcome: result of one experiment
Sample space (S): set of all possible outcomes
Sample size (n): number of trials
Not all outcomes the same due to randomness not predictable in a deterministic sense
Events
Sample space divided into events (A1, A2, A3, …)
Union and intersection of events
Disjoint and complement events

3. Mean and Variance Data: multiple measurements of same quantity
Represent data graphically using histogram
Definitions
Range:
Median: middle value when values are ordered according to magnitude
Properties
Outlier: data value that falls outside a certain number of standard deviations

4. Matlab: Histograms
>> y = [ ]’;
>> hist(y,9)  histogram plot with 9 bins
>> n = hist(y,9)  store result in vector n
>> x = [ ]’
>> n = hist(y,x)  create histogram with bin centers specified by vector x
>> mean(y)  ans =
>> var(y)  ans =
>> std(y)  ans =

5. Definition of Probability
Simple definition for finitely many equally likely outcomes
Relative frequency
General definition: P(Aj) satisfies the following axioms of probability:

6. Basic Theorems of Probability
Complementation:
Addition rule for mutually exclusive events
Addition rule for arbitrary events
Conditional probability of A2 given A1
Independent events

7. Probability Examples
Probability that at least one coin will turn heads up from five tossed coins
Number of outcomes: 25 = 32
Probability of each outcome: 1/32
Probability of no heads: P(AC) = 1/32
Probability at least one head: P(A) = 1-P(AC) = 31/32
Probability of getting an odd number or a number less than 4 from a single dice toss
Probability of odd number: P(A) = 3/6
Probability of number less than 4: P(B) = 3/6
Probability of both:
Probability of either:

8. Permutations
Permutation – arrangement of objects in a particular order
The number of permutations of n different objects taken all at a time is: n! = n
The number of permutations of n objects divided into c different classes taken all at a time is:
Number of permutations of n different objects taken k at a time is:

9. Permutation Examples Box containing 6 red and 4 blue balls
Compute probability that all red balls and then all blue balls will be removed
n1 = 6, n2 = 4
Probability
Coded telegram
Letters arranged in five-letter words: n = 26, k = 5
Total number of different words: nk = 265 = 11,881,376
Total number of different words containing each letter no more than once:

10. Combinations
Combination – selection of objects without regard to order
Binomial coefficients
Stirling formula:
Number of combinations of n different objects taken k at a time is:

11. Combination Examples Effect of repetitions
Three letters a, b, c taken two at a time (n = 3, k = 2)
Combinations without repetition
Combinations with repetitions
500 light bulbs taken 5 at a time
Repetitions not possible
Combinations

12. Matlab: Permutations & Combinations
>> perms([2 4 6])  all possible permutations of 2, 4, 6
>> randperm(6)  returns one possible permutation of 1-6
>> nchoosek(5,4)  number of combinations of 5 things taken 4 at a time without repetitions
ans = 5
>> nchoosek(2:2:10,4)  all possible combinations of 2, 4, 6, 8, 10 taken 4 at a time without repetitions

13. Matlab: Categorical Arrays
Label data in numerical arrays by category (e.g. color, size, species)
Example: flipping 10 coins
>> toss = ceil(2*rand(10,1)) randomly generate an integer = 1 or 2
10 times
>> toss = nominal(toss,{'heads','tails'})  form a categorical array
that replaces each value of 1 with ‘heads’ and 2 with ‘tails
tails heads tails heads heads heads tails heads heads heads
>> summary(toss) return the number of occurrences of each data label in the categorical array
heads tails
Repeat 100 times and plot data in histogram
>> tosses = nominal(ceil(2*rand(10,100)),{'heads','tails'})
>> T = summary(tosses)
>> hist(T’,9)