Probability Theory

1
Probability Theory

• Basic concepts
• Probability
• Permutations
• Combinations

2
Randomness

• Experiments
• 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 intersection of events
• Disjoint 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

• gtgt 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
• gtgt hist(y,9) ? histogram plot with 9 bins
• gtgt n hist(y,9) ? store result in vector n
• gtgt x 2 4 6 8
• gtgt n hist(y,x) ? create histogram with bin
  centers specified by vector x
• gtgt mean(y) ? ans 5.1589
• gtgt var(y) ? ans 6.1726
• gtgt std(y) ? ans 2.4845

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! 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

9
Permutation Examples

• Box containing 6 red 4 blue balls
• Compute probability that all red balls 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

• gtgt 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
• gtgt randperm(6) ? returns one possible permutation
  of 1-6
• 5 1 2 3 4 6
• gtgt nchoosek(5,4) ? number of combinations of 5
  things taken 4 at a time without repetitions
• ans 5
• gtgt nchoosek(2210,4) ? 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