Addition Rules for probability


Addition Rules for probability
Objective: Find the probability of compound events using the addition rules.

Many problems involve finding the probability of two or more events. For example,at
a large połitical gathering, one might wish to know, for a person selected at random, the probability that a person is female or a  Republican. In this case, there are three possibilities to consider:

 1. The person is a female.
 2. The person is a Republican.
 3. The person is both a female and a Republican.

 Consider another example.
 At the same gathering there are Republicans, Democrats, and Independents. If a person is selected at random, what is the  probability that the
person is a Democrat or an Independent? in this case, there are only two possibilities:

1. The person is a Democrat.

2. The person is an Independent.

The difference between the two examples is that in the first case, the person selectedcan be a female and a Republican at the same time. In the second case, the person selected cannot be both a Democrat and an Independent at the same time. in then second case, the two events are said to be mutually exclusive; in the first case, they are not mutually exclusive.

***********************************************************************************
Mutually exclusive: Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no Outcomes in Common).

In another situation, the events of getting a 4 and getting a 6 when a single card is drawn from a deck are mutually exclusive events, since a single card cannot be both a 4 and a 6. On the other hand, the events of getting a 4 and getting a heart on a single draw are not mutually exclusive, since one can select the 4 of hearts when drawing a single card from an ordinary deck.

***********************************************************************************
Example
Determine which events are mutually exclusive and which  are not when a single die is rolled.
a. Getting an odd number and getting an even number
b. Getting a 3 and getting an odd number
c. Getting an odd number and getting a number less than 4
d. Getting a number greater than 4 and getting a number less than 4

Solution
   a. The events are mutually exclusive, since the first event can be 1, 3, or 5,
 and the second event can be 2, 4, or 6.

b. The events are not mutualy exclusive, since the first event is a 3 and the second
can be 1,3, or 5. Hence, 3 is contained in both events.

c. The events are not mutually exclusive, since the first event can be 1,3, or 5, and
the second can be 1,2, or 3. Hence, 1 and 3 are contained in both events,

d. The events are mutually exclusive, since the first event can be 5 or 6, and the
second event can be 1, 2, or 3.


***********************************************************************************
Example 2:
Determine which events are mutually exclusive and which are not when a single card is
drawn froma deck.
a. Getting a 7 and getting a jack
b.Getting a club and getting a king
c. Getting a face card and getting an ace
d. Getting a face card and getting a spade

Solution:
Only the events in parts a and c are mutually exclusive.

***********************************************************************************


 The probability of two or more events can be determined by the addition rules. The first addition rule is used when the events are mutually exclusive.

Addition Rule 1:
When two events A and B are mutually exclusive, the probability that A or B will occur is
      
                    P(A or B)= P(A) + P(B

*********************************************************************************
Example 1:
At restaurant has 3 pieces of apple pie, 5 pieces of cherry pie, and 4 pieces of pumpkin pic in its dessert case. If a customer selects a piece of pie for dessert, find the probability that it will be either cherry or pumpkin.

Solution
Since there is a total of 12 pieces of pie, and the events are mutually exclusive,

P(cherry or pumpkin)  = P(cherry) + P(pumpkin)
                                    =  5÷12 + 4÷12    
                                        =     9÷12
                                        =     3÷4         

***********************************************************************************
Example 2:
At political rally, there 20 Republicans,  13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is either a Democrat or an  Independent.

Solution:
P(Democrat or Independent) = P(Democrat) + P(Independent)

                                               =  13÷39  +  6÷39   =  19÷39              
***********************************************************************************

Example 3:
A day of the week the is selected at random. Find the  probability that it is a weekend day.

Solution: 
    P(Saturday or Sunday)  = P(Sunday) +  P( Saturday) 

                                          =  1÷7  +   1÷7 =  2÷7



When  two events  are not mutually exclusive, we must  subtract one of the two probabilities of the outcomes that are common to both events, since they have been counted  twice. This technique is illustrated in  the below Example

***********************************************************************************
Example:
A Single card is drawn from a deck. Find the probability that it is a king or a club.

Solution:
Since the king of clubs is counted twice, one of the two probabilities must be subtracted,
as shown.

   P(king or club) = P(king) + P(club) - P(king of clubs)
                            =   4÷52  +  13÷52 -  1÷52
                           =         16÷52
                           =           4÷13
***********************************************************************************

When events are not mutually exclusive, addition rule 2 can be used to find the
probability of the events

Addition Rule 2:

lf A and B are not mutually exclusive, then

  P(A or B)= P(A) + P(B) - P(A and B)

Note: This rule can also be used when the events are mutually exclusive, since P(A and B) will always equal 0. However, it is important to make a distinction between the two situations.

***********************************************************************************
Example 1:
In a hospital unit there are eight nurses and five physicians. Seven nurses and three physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

Solution:
The sample space is shown.

Staff                            Females                               Males                               Total
*****************************************************************
Nurses                             7                                        1                                        8

Physicians                       3                                        2                                        5

Total                               10                                       3                                       13

The probability is

             P(nurse or male) =  P(nurse) + P(male) - P(male nurse)
                                        =   8÷13 +  3÷13 -  1÷13 
                                           
                                        =   10÷13  

***********************************************************************************

 Example 2 :
On New Year's Eve, the probability of a person driving while intoxicated is 0.32, the probability of a person having a driving accident is 0.09, and the probability of a person having a driving accident while intoxicated is 0.06. What is the probability of a person driving while intoxicated or having a driving accident?

Solution:

        P(intoxicatcd or accident) = P(intoxicated) + P(accident) - Pintoxicated and accident)
   
                                                  =  0.32 + 0.09 - 0.06
                                                
                                                  =   0.35

***********************************************************************************
For Three Events:
The probability rules can be extended to three or more events. For three mutually exclusive events, A, B, and C,
                   P(A or B or C) = P(A) + P(B) + P(C)


For three events that are not mutually exclusive,
   P(A or B or C) = P(A) + P(B) + P(C) - P(A and B)- P(A and C) - P(B and C) + P(A and B and C)






Thanks for Visiting Here                                             
     
    A R Statistics





Comments

Post a Comment

Popular posts from this blog

Theory of Attributes- Basic concept and their applications:

Image
Theory of Attributes- Basic concept and their applications: ATTRIBUTES:       Literally, an attribute means a quality or characteristic.Theory of attributes deals with qualitative characteristics  which not amenable to  quantitative measurements  are and hence need slightly different statistical treatment from that of the  variables. Examples of attributes are drinking,smoking, blindness, health, honesty, etc. An attribute may be marked by its presence (possession) or absence (dispossession) in  member of given population. It may be pointed out that the methods of statistical analysis applicable to the study of variables can also be used to a great extent in the theory of attributes and vice-versa. For example, the presence or absence of an attribute  may be regarded as changes in the values of a variable which can possess only two values, viz., 0 and 1. NOTATIONS: Suppose the population is divided into two classes according the to the  presence or absence of a a single attr
Read more

MCQ's OF VITAL STATISTICS

Image
Q.1 Vital statistics is mainly concerned with : (a) births (b) deaths (c) marriages (d) all the above Q.2 Population statistics mainly display the records the pertaining to: (a) foetal deaths (b) population of regions (c) morbidity (d) all the above Q.3 Complete count of the heads of  people of a country is known as: (a) census (b) vital statistics (c) demography (d) none of the above Q. 4 Vital statistics throws light on: (a) changing pattern of the population during intercensal period (b) virility of races (c) growth of population (d) all the above Q.5 The registration of births, deaths and marriages are: (a) a fancy of society (b) a part of medical research (c) a legal document (d) all the above Q.6 Vital statistics is greatly utilised by: (a) acturies (b) planners (c) social reformers (d) all the above Q.7 In India, the collection of vital statistics started for the first times in: (a)  720 (b) 1886 (c) 1969 (d) 1946 Q. 8 The registrat
Read more

A.R Statistics

Image
                                    Historical Note of Statistics       ‘Statistics’ is derived from the Latin word ‘status’ which means a political state. This suggests that statistics is as old as human civilisation. In the year 3050 B.C., perhaps the first census was held in Egypt. In India also, about 2000 years ago, we had an efficient system of collecting administrative statistics, particularly, during the regime of Chandra Gupta Maurya (324-300 B.C.). The system of collecting data related to births and deaths is mentioned in Kautilya’s Arthshastra (around 300 B.C.) A detailed account of administrative surveys conducted during Akbar’s regime is given in Ain-I-Akbari written by Abul Fazl. 2019-20        Captain John Graunt of London (1620-1674) is known as father of vital statistics due to his studies on statistics of births and deaths. Jacob Bernoulli (1654-1705) stated the Law of Large numbers in his book “Ars Conjectandi’, published in 1713.
Read more