The file is related to the van diagram and probability distribution

1. Biostatistics
Lecture No 17

2. Biostatistics
Venn Diagram is a graphical representation of sample space and events. It represent events as circles
enclosed in a rectangle. The rectangle represents the sample space and each circle represents an event.
•Unions of Two Events
If A and B are two events, then the union of A and B, denoted by , represents the event composed of all
basic outcomes in A or B. Union means the set of all elements that belongs to at least one of the sets A and
B.
Venn diagram
A B

{ | }
A B x x A or x B
   

3. Biostatistics
{ | and }
A B x x A x B
   
• Intersections of Two Events
If A and B are events, then the intersection of A and B, denoted by , represents the event composed of all
basic outcomes in A and B.”
A B

Complement of an Event
Let S denotes the sample space of a probability experiment and let A denote an event, then complement of event
A is denoted by which contains all simple points in the sample space S that are not simple event A. If A
represent any event and represents the complement of event A, then
A
S
( ) ( ) ( )
( ) 1 ( )
A S A
P A P S P A
P A P A
 
 
 
A
A

4. Biostatistics
Example
According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is
the probability that a randomly selected household does not own a dog?
( ) 31.6%, ( ) 1- ( )
( ) 1-31.6% ( ) 68.4%
E Own a dog
P E P E P E
P E and P E

 
 

5. Biostatistics
 Two event say A and B from a single experiment are called mutually exclusive if and only if they cannot
occur at a time. In another words they have no point in common.
Fore example a single birth must be either male or female baby. Three or more events generated from the same
experiment are mutually exclusive pairwise.
 If two events are mutually exclusive then . .For three mutually exclusive events .
Suppose there is a sample space
Here , thus event “A” and “B” are mutually exclusive events.
A B
   A B C
   
{1,2,3,4,5,6}
{ numbers} {1,3,5},
{ numbers} {2,4,6}
S
A odd
B even

 
 
A B
  
A
S
B
A B
  
Types of Event
Mutually Exclusive events

6. Biostatistics
If the two event can occur at the same time they are not mutually exclusive. Fore example, when we draw a
card from a deck of 52 cards, it can be both a king and diamond. Therefor king and diamond are not mutually
exclusive. .
The events are called not mutually exclusive if they have at least one outcome in common. If two events “A”
and “B” are not mutually exclusive, then .
Consider the following sample space
Here , thus event “A” and “B” are not mutually exclusive events.
A B
  
{1,2,3,4,5,6},
{ numbers} {1,3,5}, {prime number} {2,3,5}
S define
A odd B

   
A B
  
A
S
B
A B
  
Not Mutually Exclusive events

7. Biostatistics
Events are said to be collective exhaustive when the union of mutually exclusive event constitute the sample
space itself. For example when we toss a coin the outcome Head and Tail are exhaustive because the union
constitute the whole sample space.
 When we toss a coin, then
Exhaustive Events
{1,2,3,4,5,6}
Let A={1,2}, B={3,4,5}, C={6}
S 
Here the events “A”, “B” and “C” are mutually exclusive because and Exhaustive because of
A B C
   
A B C S
  
Two events A and B are said to be equally likely, when one event is as likely to occur as the other. For example
when a fair die is tossed, the head is as likely to appear as the tail.
Equally likely events

8. Biostatistics
Example: A class of 60 students has 30 first divisioners, 20 second and 10 third divisioners. One student is
selected at random find the probability that the selected student is i) first divisioner ii) first or second divisioner
ii)second or third divisioner.
Solution: There are total 60 students thus there are 60 possible outcomes in “S”
i) Let “A” be the event that the selected student is first divisioner, therefore there are 30 outcomes favorable
to the event “A”
ii) Let “B” be the event the selected student is first or second divisioner, therefor there are 50 students, therefor
iii) Let “C” be the event that the selected student is the second or third student, therefore
( ) 30 1
( )
( ) 60 2
n A
P A
n S
  
( ) 50 5
( )
( ) 60 6
n B
P B
n S
  
( ) 30 1
( )
( ) 60 2
n C
P C
n S
  

9. Biostatistics
Example: A fair die is thrown , find the probability that the upper face on the die is i) maximum ii) prime number
iii) multiple of 3 iv) multiple of 7.
Example: Two dice are rolled . Let “A” be the event that both face are same, “B” be the event that total on the
two dice is less than 5 and “C” be the event there is at least one ace. Write the sample space S and find the
probabilities of the events defined above.
Example: Suppose a bag contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue
candies, and 2 green candies. Suppose that a candy is randomly selected.
(a) What is the probability that it is brown?
(b) What is the probability that it is blue?