Probability and Discrete Events

In Chapter 3, the author introduces the concept of probability, the sample space, interpretations of probability, conditional probability.

If we were to provide a general definition of probability, we might say that is the relative likelihood of obtaining some partiuclar data or combination of data. In turn, we might define statistics as the use of probability techniques to make intelligent inferences regading a population on the basis of sample data.

Thus, statistics is based upon the concept of probability.

Probability

When we cite the probability of an event, we are making reference to the chance that a particular outcome will occur in comparison to the total number of outcomes that are possible. For instance, if we toss a coin, the two possible outcomes are heads and tails. Thus, the probability of getting a head on a single coin toss is one out of two, or fifty percent. If we toss two coins, there are four possible outcomes:

  • head/head
  • head/tail
  • tail/head
  • tail/tail
Consequently, the probability of getting a head and a tail in the toss of two coins is two out of four (50%), while the probability of getting two tails is one out of four (25%).

In making a determination of probability, it is necessary to assess the total number of outcomes that are possible. To do this, we will refer to the Rules of Counting (ref. F. H. Zuwaylif, General Applied Statistics):

Rule 1

If outcome "A" can occur in m ways, and outcome "B" can occur in n ways, then outcome "A and B" can occur in (m)(n) ways.

Example: Roll a die and toss a coin.

  • The die has six possible outcomes
  • The coin has two possible outcomes
  • The total number of outcomes is (6)(2)=12
  • P(2 and tail)=1/12=0.0833

Rule 2

In permutations, the order of the outcomes matters. There are two permutations of the outcomes "A" and "B": AB and BA.

There are n! permutations of n outcomes or objects.

The quantity n! is read n-factorial and is calculated as:

n!=(n)(n-1)(n-2)(n-3) . . . (1)

So, 5!=(5)(4)(3)(2)(1)=120

Example: There are 6!=720 ways for six people to stand in a line.

Rule 3

There are n!/(n-r)! permutations of n objects or outcomes taken in groups of r.

Example: The number of permutations of the letters "A", "B", "C", and "D" taken two at a time is

(4!)/(4-2)!=24/2=12

where the permutations are AB,AC,AD,BC,BD,BA,CD,CA,CB,DA,DB, and DC.

Rule 4

In combinations, the order of outcomes or objects does not matter. For example, there is only one combination of the letters "A" and "B".

There are n!/((n-r)(r!)) combinations of n objects or outcomes taken in groups of r.

Example: The number of combinations of or the letters "A", "B", "C", and "D" taken two at a time is

(4!)/((4-2)!(2!))=24/(2)(2)=6

where the combinations are AB, AC, AD, BC, BD, and CD.

Rule 5

Consider a total of n outcomes or objects, of which r1 are of one type, r2 are of a second type, r3 are of a third type, etc.

The number of permutations is n!/((r1!)(r2!)(r3!). . . )

Example: There would be

3!/((1!)(2!))=3

permutations of the letters "A", "B", and "B": ABB, BAB, and BBA.

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