In the previous page, we discussed about the concept "Probability". Now we find a more precise definition of this concept.

Realizing a statistical experiment with `n` trials and event E occurs `f`_E times (`f`_E is frequency of event E). We already know that the ratio between `f`_E and `n` is relative frequency of E.

Probability of event E, symbolized as `P("E")`, is defined as the limit of relative frequency when `n` approach infinite:

`P("E")=lim_(n rarr oo) f_"E"/n`

(1)

 

From the definition, we can determine some characteristics of probability:

• `0<=P("E")<=1`
• If E_`i` is elementary event then:
  

  `sum_(i=1)^k P("E"_i)=1`

  (2)

Equally likely outcomes

There are cases when the ability of all the outcomes are similar. In such cases, the probability of all elementary events are equal. If the sample space consists of `N` equally likely outcomes `"E"_i`, from (2) we can conclude that:

`P("E"_i)=1/N`

(3)

In this case, if event E consists of `f` elementary events, then:

`P("E")=f/N`

(4)

Formula (4) is used frequently to calculate probability.

Example 9 : Given Example 1, if the coin is absolutely homogeneous, A is the event which upper face is H, then `P("A")=0,5`

Example 10 : Given Example 2, if the die is absolutely homogeneous:

• E_`i` : the event which upper face is `i` , then `P("E"_4) = 1/6`
• A : the event which upper face is even, then `P("A") = 3/6 = 0,5`

Example 11 : Given Example 4, if both the dice are absolutely homogeneous, `"A"_i is the event which the sum of two upper faces equal to `i`. Then A₆ is the set consists of 5 elements:

  A₆ = { (15), (24), (33), (42), (51) }

  So   `P("A"_6)=5/36`

 

To determine the probability of an event, we have to consider the situation, then find the appropriate method to determine. We can refer to the following cases.

• If event E cannot occur, then `P("E")=0`. This event can be symbolized as `O/`.
• If event E is certain, then `P("E")=1`.
• The complementary event of event E, symbolized as E^c, consists of all the outcomes in the sample space that are not included in event E (Fig. 1).
  
  

  Fig. 1 Complementary event

  Then   `P("E"^"c")=1-P("E")`

• The union of two events E and F, symbolized as E∪F, consists of all the outcomes contained in either E or F (Fig. 2a).
• The intersection of two events E and F, symbolized as E∩F, consists of all the outcomes contained in both E and F (Fig. 2b).
  
  

  Fig. 2 Union of events (2a) and intersection of events (2b)

• The relation of probability of these events:
  

  `P("E"uu"B")=P("E")+P("F")-P("E"nn"F")`(5)

  Formula (5) is referred as addition rule of probability

• E and F are mutually exclusive if they can not happen at the same time, so they do not have any outcome in common (Fig. 3).
  
  

  Fig. 3 Mutually exclusive events

    Then :   E∩F = ∅   and   `P("E"nn"F")=0`

• E is a subset of F, symbolized as E ⊂ F, when all the outcomes of E are also the outcomes of F (Fig. 4).
  
  

  Fig. 4 Subset event

    Then :   `P("E")<=P("F")`