PROBABILITY MEASURES AND DISTRIBUTIONS

The following tools are the keys to understanding probability:


DEFINITIONS:


COUNTING METHODS:

Note that counts are not numerical data.

The following are methods of counting the numbers of items and outcomes in everyday life:

A note on the values used above:

Examples of each kind of experiment:


OBJECTS
REPLACED
OBJECTS NOT
REPLACED
ORDER
MATTERS
Power Principle

nr

Permutation

P(n,r)

NO
ORDER
Recombination

R(n,r)

Combination

C(n,r)


PROBABILITY MEASURES:

The following are probability measures that occur in everyday life.


REPETITION PROBABILITY:

Note that the following distributions use categorical data:


RANDOM VARIABLE:

EXPECTED VALUE:

Appendix

OBTAINING EQUALLY LIKELY OUTCOMES WITH A RECOMBINATION SCENARIO

A sample case: How many colors can a light panel show by changing lightbulbs?

Why Conventional Methods Do Not Provide Equally Likely Outcomes:

Method 1: Prepared Cards (or balls)

The procedure is:

  1. Prepare one card or ball for each outcome in a table of all possible outcomes, made as above (this example has 28).
  2. Choose one card to pick the entire combination.

Some advantages:

Some disadvantages:

Method 2: The gap method

The procedure

  1. Make a rule for the ordering of the choices (alphabetic order, numeric order, spectral order, etc - spectral in this example).
  2. Prepare r choice balls (or cards) for each available choice in the n choices (6 of each of 3 colors in this example).
  3. Place one of each kind of choice ball (or card) in a bin (3 in this example). Keep the others in a box for later.
  4. Also prepare r-1 order balls (or cards) numbered 1 through r-1 (1 to 5 in the example). Use letters if the original choices are numbers).
  5. Place all of the order balls in the bin.
  6. Simultaneously draw r balls (6 in the example) from the bin.
  7. Sort any choice balls drawn according to the rule made earlier and place them in a tray in sorted order.
  8. Sort any order balls drawn in order, and place them in a tray in sorted order to the right of the choice balls.
  9. Note which of the order balls are missing from the rack. Each one is a gap.

    The number of gaps is always one less than the number of choice balls drawn.

  10. Leaving the first choice ball where it is, place the other choice balls into the gaps. Keep the choice balls in order.

      Note: a gap can appear at either one or both ends of the choice levels.

  11. Replace each order ball with a choice ball (from the box) identical to the choice ball to its left.

The balls in the tray now contain the desired choice. All outcomes are equally likely.

Using a ball set R G B 1 2 3 4 5 with "$" as a gap.
Examples:

DrawingGap FindingMove ChoicesFinal Outcome
G12345→ G12345→ G12345→ GGGGGG - green
GB1245→ GB12$45→ G12B45→ GGGBBB - cyan
RG1234→ RG1234$→ R1234G→ RRRRRG - vermillion
RG2345→ RG$2345→ RG2345→ RGGGGG - leaf
RGB123→ RGB123$$→ R123GB→ RRRRGB - pink
RGB135→ RGB1$3$5→ R1G3B5→ RRGGBB - white
RGB245→ RGB$2$45→ RG2B45→ RGGBBB - light azure
RGB345→ RGB$$345→ RGB345→ RGBBBB - light blue
RGB145→ RGB1$$45→ R1GB45→ RRGBBB - lilac
RGB124→ RGB12$4$→ R12G4B→ RRRGGB - peach

Some advantages:

Some disadvantages:

Links: