Thursday, October 30, 2008

Time For Self Evaluation

Directions: Read each question below and tick mark the correct option.


1. Which of the following is the sample space when 2 coins are tossed?
  
 
 {H, T, H, T}
 {H, T}
 {HH, HT, TH, TT}
 None of the above.




2. At Kennedy Middle School, 3 out of 5 students make honor roll. What is the probability that a student does not make honor roll?
  
 
65%
40%
60%
None of the above.




3. A large basket of fruit contains 3 oranges, 2 apples and 5 bananas. If a piece of fruit is chosen at random, what is the probability of getting an orange or a banana?
  
 






None of the above.




4. A pair of dice is rolled. What is the probability of getting a sum of 2?
  
 






None of the above.



5. In a class of 30 students, there are 17 girls and 13 boys. Five are A students, and three of these students are girls. If a student is chosen at random, what is the probability of choosing a girl or an A student?
  
 






None of the above.




6. In the United States, 43% of people wear a seat belt while driving. If two people are chosen at random, what is the probability that both of them wear a seat belt?
  
 
86%
18%
57%
None of the above.



7. Three cards are chosen at random from a deck without replacement. What is the probability of getting a jack, a ten and a nine in order?
  
 






None of the above.



8. A city survey found that 47% of teenagers have a part time job. The same survey found that 78% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part time job and plans to attend college?
  
 
60%
63%
37%
None of the above.




9. In a school, 14% of students take drama and computer classes, and 67% take drama class. What is the probability that a student takes computer class given that the student takes drama class?
  
 
81%
21%
53%
None of the above.




10. In a shipment of 100 televisions, 6 are defective. If a person buys two televisions from that shipment, what is the probability that both are defective?
  
 






None of the above.

11. If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8?
  
 


1

0

None of the above.




                                                                                                               

Wednesday, October 29, 2008

Some Important Results

                                        _       
# We have P(A) = 1 -  P(A),
    therefore, P(A)  1.           _
                                 {Since P(A) ≥ 0,}    
                                    _
# P(Φ) = 0, since Φ = (S) and 
                  _
   P(Φ) = P(S) = 1- P(S) = 1-1 =

# If in an experiment with n equally likely (and exhaustive) outcomes, m outcomes are favourable to an event A,      then probability of an event A is defined simply as 
                  P(A) = The number of possible outcomes of count as A 
                                   The total number of possible outcomes    
# For every event A in S, P(A) 0.

#  For every event A in S, P(A) 1.

# Probability of an impossible event is always zero i.e.,
    P(Φ) = 0
# For the sure event or certain event, P(S) = 1. 

ILLUSTRATIONS
Experiment 1:  A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a king?  [IMAGE]
Probability:  
P(not king) = 1 - P(king)
 
 = 1 -  4 
52
 
 = 48
52
 
 = 12
13

Experiment 2:  A single 6-sided die is rolled. What is the probability of rolling a number that is not 4?  dice


Probability:  
P(not 4) = 1 - P(4)
 
 = 1 - 1
6
 
 = 5
6

Experiment 3:  A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a club?  [IMAGE]
Probability:  
P(not club) = 1 - P(club)
 
 = 1 - 13
52
 
 = 39
52
 
 = 3
4

Experiment 4:  A glass jar contains 20 red marbles. If a marble is chosen at random from the jar, what is the probability that it is not red?  [IMAGE]
Probability:  
P(not red) = 1 - P(red)
 
 = 1 - 1
 
 = 0
 
Note: This is an impossible event.
Experiment 6:  A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner?  spinner
Sample Space:  {yellow, blue, green, red}
Probability:  
P(not red) = 1 - P(red)
 
 = 1 - 1
4
 
 = 3
4

Summary:  The probability of an event is the measure of the chance that the event will occur as a result of the experiment. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:
  • If P(A) > P(B) then event A is more likely to occur than event B.
  • If P(A) = P(B) then events A and B are equally likely to occur.
  • If event A is impossible, then P(A) = 0.
  • If event A is certain, then P(A) = 1.
  • The complement of event A is .   
  •  P() = 1 - P(A)


Probability Of An Event

RESULT 1
Consider the set
S of all possible outcomes of an experiment or a trial. A set of this kind sometimes called a Probability space or Sample space. Any event A can be represented by the subset of S, which contains all the outcomes in which the event occurs

Let S contains a finite number of equally likely outcomes, say m, so that n(S) = n. Let the event A has m sample points so that n(A) = m.
We have
       P(A) = n(A)/ n(S) = m/n
                  = Number of Favourable Outcomes
                      Number of Possible Outcomes
                                        
              Since A is a subset of S,

          
   Therefore 0 n  i.e., 0 ≤ m/n ≤ 1
                                            Hence ≤ P(A) ≤ 1
Note: The probability of an event A is a number between 0 and 1 inclusive. If  P(A) = 0, then the event cannot possibly occur.
If  P(A) = 1, then the event is certain to occur.
RESULT 2       
       _
Let  A denote the event  'A does not occur
                              _           _
                 Now P(A)  = n(A) /n(S) = n-m = 1-m     
                                                                  n           n
                                                                 = 1 -  P(A)
                                                 _       
                             Therefore P(A) = 1 -  P(A)
                                                              _
                            This implies P(A) + P(A) = 1                             


Friday, October 24, 2008

Classification Of Events

TYPES OF EVENTS

As explained earlier :
An event is a collection of one or more of the outcomes of an experiment
OR 
In simple words,
An event is a subset of a sample space

An event can be classified into various types as follows :-

(1)  SIMPLE EVENT :- 

A simple event has only one sample point of the sample space. Each of the final outcomes from experiment is called a simple event. In other words, a simple event includes one and only one outcomes.

For example, consider the experiment of rolling a die once, therefore sample space S is given by
S= {1, 2, 3, 4, 5, 6}.
Now event like Getting a ‘3’ , Getting ‘5’, Getting ‘6’ are simple events.

A Simple Event is also called an Elementary Event.


(2) COMPOUND EVENT :-

A compound event is a collection of more than one outcome of an experiment.

For example, consider the experiment of rolling a die once, therefore sample space S is given by
S= {1, 2, 3, 4, 5, 6}. Then, events like
Getting an even number.
(Possible outcomes :- 2, 4, 6)
                   or
Getting a number ≥ 3.
(Possible outcomes :- 3, 4, 5, 6) are compound events
Compound Event is also called a Composite Event.

(3) IMPOSSIBLE EVENT :- 

Every non empty subset A of a sample space S , is called an Event. The empty set Φ is also a subset of the sample space S, therefore, it also represents an event. Now consider an experiment in which two dice are thrown. The sample space S of this experiment consists of 36 points


S = {(1, 1), (1, 2),….......…, (1, 6),
         (2, 1), (2, 2),……….…, (2, 6),
         (3, 1),…………….....…., (3, 6),
         ........………………………………,
        (6, 1),….……………....., (6, 6) }.

If A is an event such that,
A= the event that the sum of the numbers on the faces is greater than 12, then, A= Φ

Since no outcome of this experiment is a member of Φ , the event represented by Φ cannot occur at all. We call the event Φ as an Impossible Event. Thus,

An event corresponding to the empty set is called an impossible event or null event

(4) SURE EVENT :- 

Consider the experiment of rolling a die once.

Here Sample Space S is given by
S = {1, 2, 3, 4, 5, 6}

Now let A be the event 'the number turns up is even or odd'.

So, here the event A is that the number on the face is one of the numbers 1, 2, 3, 4, 5, 6, then

A = {1, 2, 3, 4, 5, 6}
Here A is a Sure (or Certain) Event.
Thus
The event corresponding to the entire Sample Space is called a Sure (or Certain) Event.

(5) MUTUALLY EXCLUSIVE EVENTS :- 

Consider the eaperiment of throwing a dice.

Let A be the event, 'the number obtained is less than 4 '. Then,
A = {1, 2, 3} .

Let B be the event, 'the number obtained is atleast 5 '. Then,
B = {5, 6}.

Clearly A ∩ B = Φ.

Thus the joint occurance of A and B is thus an impossible event.
The events A & B are called Mutually Exclusive Events

Thus, "Two events A and B are called mutually exclusive events if the occurance of any one of them excludes the occurance of the other event i.e. if they cannot occur simultaneously"

In terms of sets,Two events A and B are called mutually exclusive events if            A ∩ B = Φ and n(A ∩ B) = 0.

(6) EXHAUSTIVE EVENTS:-

If two events A and B are such that A U B = S then P(A U B) = 1 and the events A and B are said to be Exhaustive.

For Example. 
Let S be the sample space when an ordinary die is thrown.
If A is the event the number is less than 5 and B the event the number is greater than 3, then the events A and B are exhaustive as A U B = S.

Thus,
If E1 , E2 .........., En are the subsets of a sample space S, and 
if E1 U E2 U E3 U .....U Em = S, then E1, E2,..........En form a set of Exhaustive Events.

EVENT & SAMPLE SPACE OF A RANDOM EXPERIMENT


Event associated with a Random Experiment 

An event associated with a Random Experiment is a Situation or a result which is based on the possible outcomes of the experiment.


For example, we toss a fair and balanced dice with its six faces marked with dots from one to six.
We find one face out of the six faces at the top. We count the number of dots on this face. 

The outcome is ‘n’, if the number of dots on the top are n = 1, 2, 3, 4, 5, 6. Thus, in this random experiment, the possible outcomes are1, 2, 3, 4, 5, 6.

The following events are associated with this Random Experiment:-

Getting the number 1 on the top face.
Getting the number 2 on the top face
Getting the number 3 on the top face

Getting the number 4 on the top face
Getting the number 5 on the top face
Getting the number 6 on the top face

All these six events are called elementary events as they are associated with the occurrence of single outcome in each case. 

However there can be many compound events (which are associated with more than one outcome) associated with the Random Experiment as below:-


Getting a number ≥ 3.
Here, the four outcomes 3, 4, 5, 6 favour the event.


Getting an even number.
Here, the three outcomes 2, 4, 6 favour the event.

Many more such compound events can be stated.

Note:- Events associated with Random Experiment are denoted by capital letters 

SAMPLE  SPACE

The set of all possible outcomes of a random experiment is called the Sample Space associated with it and it is generally denoted by S.

If E1 , E2 , E3 ,……..,En are the possible outcomes (or elementary events) of a random experiment , then ,
S = { E1 , E2 , E3 ,……..,En } is the sample space associated to it. 

I l l u s t r a t i o n 1

Consider the experiment of tossing two coins together or a coin twice. In this experiment the possible outcomes are:


  • Head on first and Head on second,
  • Head on first and Tail on second,
  • Tail on first and Head on second,
  • Tail on first and Tail on second,

If we define:

HH = Getting Head on both coins,
HT = Getting Head on first and Tail on second,
TH = Getting Tail on first and Head on second,
TT = Getting Tail on both coins.

Then,
HH, HT, TH and TT are the elementary events associated to the random experiment of tossing of two coins. The sample space associated to this experiment is given by 

S = {HH, HT, TH, TT}

I l l u s t r a t i o n 2

Consider the experiment of tossing two dice together or a die tossed twice. 


If we define:

Eij = Getting a number i on the upper face of first die and the number j on the upper face of second die,
Where i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6.

Then, Eij are the elementary events associated to 
the random experiment of tossing of two dice and are generally 
denoted by (i, j)

Thus, (1, 1), (1, 2),……, (1, 6), (2, 1), (2, 2),……….…, (2, 6),
           (3, 1),……………...…., (3, 6), ………………………..………,
            .………………………………,(6, 1),…………….…...., (6, 6), 
are 36 elementary events associated to the random experiment of tossing of two dice and the sample space associated to it is given by

S =    { (1, 1), (1, 2),…........…, (1, 6),
             (2, 1), (2, 2),……….…, (2, 6),
             (3, 1),…………….....…., (3, 6),
             ………………........………………,
             …………………………........……,
             (6, 1),…………...……...., (6, 6) }.