Probability Theory Basic concepts Probability Permutations Combinations

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

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

Matlab: Histograms >> y = [1 3 5 8 2 4 6 7 8 3 2 9 4 3 6 7 4 1 5 3 5 8 9 6 2 4 6 1 5 6 9 8 7 5 3 4 5 2 9 6 5 9 4 1 6 7 8 5 4 2 9 6 7 9 2 5 3 1 9 6 8 4 3 6 7 9 1 3 4 7 5 2 9 8 5 7 4 5 4 3 6 7 9 3 1 6 9 5 6 7 3 2 1 5 7 8 5 3 1 9 7 5 3 4 7 9 1]’; >> hist(y,9)  histogram plot with 9 bins >> n = hist(y,9)  store result in vector n >> x = [2 4 6 8]’ >> n = hist(y,x)  create histogram with bin centers specified by vector x >> mean(y)  ans = 5.1589 >> var(y)  ans = 6.1726 >> std(y)  ans = 2.4845

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

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

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:

Permutations Permutation – arrangement of objects in a particular order The number of permutations of n different objects taken all at a time is: n! = 1.2.3. . .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:

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:

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:

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

Matlab: Permutations & Combinations >> perms([2 4 6])  all possible permutations of 2, 4, 6 6 4 2 6 2 4 4 6 2 4 2 6 2 4 6 2 6 4 >> randperm(6)  returns one possible permutation of 1-6 5 1 2 3 4 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 2 4 6 8 2 4 6 10 2 4 8 10 2 6 8 10 4 6 8 10

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 2 1 2 1 1 1 2 1 1 1 >> 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 7 3 Repeat 100 times and plot data in histogram >> tosses = nominal(ceil(2*rand(10,100)),{'heads','tails'}) >> T = summary(tosses) >> hist(T’,9)