Probability Part II

    

Probability:
                    Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of chance, such as card games, slot machines, or lotteries. In addition to being used in games of chance, probability theory is used in the fields of insurance, investments, and weather forecasting, and in various other areas.
                 Probability is the basis of inferential statistics. For example, predictions are based on probability, and hypotheses are tested by using probability.

       The basic concepts of probability are explained in this lecture. These concepts in-clude probability experiments, sample spaces, the addition and multiplication rules, and
the probabilities of complementary events.

The theory of probability grew out of the study of various games of chance using coins, dice, and cards. Since these devices lend themselves well to the application of concepts of probability, they will be used in this as examples. This section begins by explaining some basic concepts of probability. Then the types of probability and probability rules are discussed.
            Processes such as flipping a coin, rolling a die, or drawing a card from a deck are called probability experiments.

probability experiment: A probability experiment is a chance process that leads to well-defined results called outcomes.

Outcome: An outcome is the result of a single trial of a probability experiment.

A trial means flipping a coin once, rolling one die once, or the like.
 When a coin is tossed, there are two possible outcomes: head or tail. (Note: We exclude the possibility of a coin landing on its edge.) In the roll of a single die, there are six possible outcomes:
1, 2, 3,4, 5, or 6.

Sample Space:In any experiment, the set of all possible outcomes is called the sample space.
A sample space is the set of all possible outcomes of a probability experiment.
Some sample spaces for various probability experiments are shown here.

Experiment                                                      Sample space

Toss one coin                                                     Head, tail
Roll a die                                                         1,2, 3, 4, 5, 6
Answer a true-false question                            True, false
Toss two coins                                   Head-head, tail-tail, head-tail, tail-head

        It is important to realize that when two coins are tossed, there are four possible outcomes, as shown in the fourth experiment above. Both coins could fall heads up. Both coins could fall tails up.
Coin 1 could fall heads up and coin 2 tails up. Or coin 1 could fall tails up and coin 2 heads up.

    Heads and Tails will be abbreviated as H and T.

How to Find Sample Spaces,
Example 1:
Find the sample space for rolling two dice.
Solution:
Since each die can land in six different ways, and two dice are rolled, the sample space
can be presented by a rectangular array, as shown in Figure 1. The sample space is the
list of pairs of numbers in the chart.

	Die 2
Die 1	1               2             3             4                  5                    6
1 2 3 4 5 6	(1,1        (1.2)      (1,3)      (1.4)          (1,5)             (1,6) (2, 1)     (2, 2)      (2,3)     ( 2,4)          (2,5)             (2, 6) (3,1)      (3, 2)      (3, 3)     (3,4)          (3,5)             (3, 6) (4,1)       (4,2)       (4,3)     (4,4)          (4,5)             (4,6) (5,1)       (5,2        (5.3)      (5,4)          (5,5)             (5,6) (6,1)       (6,2)       (6,3)      (6,4)          (6,5)             (6,6)

                                                                                Figure 1

Example 2:
Find the sample space for drawing one card from an ordinary deck of cards.

Solution:
Since there are four suits (hearts, clubs, diamonds, and spades) and 13 cards for each
suit (ace through king), there are 52 outcomes in the sample space. See Figure 2.

Sample Space of Drawing a Card

Example 3:
Find the sample space for the gender of the children if a family has three children.
 Use B for boy and G for girl.

Solution
There are two genders, male and female, and each child could be either gender. Hence,
there are eight possibilities, as shown here.
BBB    BBG     BGB     GBB     GGG     GGB     GBG      BGG       { (2ⁿ)= 2³}

   Event:
     An outcome was defined previously as the result of a single trial of a probability experiment.
 In many problems, one must find the probability of two or more out comes. For this reason, it is necessary to distinguish between an outcome and an event.

 An event consists of a set of outcomes of a probability experiment

An event can be one outcome or more than one outcome.
For example, if a die is rolled and a 6 shows, this result is called an outcome, since it is a result of a single trial. 

Simple event:  An event with one outcome is called a simple event.

Compound event: The event of getting an odd number when a die is rolled is called a compound event, since it consists of three outcomes or three simple events.
In general, a compound event consists of two or more outcomes or simple events.

There are three basic interpretations of probability:

1. Classical probability

2. Empirical or relative frequency probability

3. Subjective probability

1. Classical probability:  Classical  probability uses sample spaces to determine  numerical probability that an event will happen. One does not actually have to perform the experiment to determine the probability.Classical by probability is so named  because it was the first type of probability studied. formally by  mathematicianin the 17th and the 18thcenturies.
      Classical probability assumes that all outcomes in the sample space are equally likely to occur. For example, when a single die is rolled, each outcome has the same probability of occurring. Since there are six outcomes, each outcome has a probability of 1/6. When a card is selected from an ordinary deck of 52 cards, one assumes that the deck has been shuffled, and each card has the same probability of being selected. In this case, it is 1/52

Equally Likely Events: Equally likely events are events that have the same probability of occurring.

Formula for Classical Probability

The probability of any event E is
 number of outcomes in E divide by total number of outcomes in the sample space

This probability is denoted by

              P (E) = n (E)/n(S)

This probability is called classical probability, and it uses the sample space S.

The formula shows that when finding classical probabilities, count the number of out

comes in the sample space for the denominator, and then count the number of outcomes

in the event under study for the numerator.

Probabilities can be expressed as fractions, decimals, or — where appropriate

percentages.
 If one asks, " What is the probability of getting a head when a coin is

tossed?" typical responses can be any of these three.

 " One-half."

 " Point five."

 " Fifty percent.

Remark:Strictly speaking, a percent is not a probability. However, in everyday language, probabilities are often expressed as percents (i.e., there is a 60 % chance of rain tomorrow.).
 For this reason, some probabilities will be expressed as percents in this context.

Examples on Classical Probability

1). For a card drawn from an ordinary deck, find the probability of getting a king.

Solution
Since there are 52 cards in a deck and there are 4 kings,
P (king) = 4÷52                          {P(E)=m÷n} where, m is Favourable cases
           = 1÷13                                                          n is total Exhaustive cases

2). If a family has three children, find the probability that all the children are girls.

Solution
The sample space for the gender of children for a family that has three children is

BBB, BBG, BGB, GBB, GGG, GGB, GBG, and BGG . (2³)

Since there is one way in eight possibilities for all three children to be girls,

 i.e. P(probability that all the children are girls) =  P (GGG) = 1÷8               {P(E)=m÷n}


3). A card is drawn from an ordinary deck. Find these probabilities.
a. Of getting a jack
b. Of getting the 6 of clubs
c. Of getting a 3 or a diamond

Solution:
a. Of getting a jack
Total number of Cards = 52
Number of Jack Cards = 4

It means  there are 4 outcomes in event E and 52 possible outcomes in the sample space.
 Hence,
P (jack) = 4÷52  =1÷13                                        {P(E)=m÷n}


b. Of getting the 6 of clubs

Since there is only one 6 of clubs in event E, the probability of getting a 6 of clubs is
P (6 of clubs) =1÷ 52


c. Of getting a 3 or a diamond

 There are Four 3s and 13 diamonds, but the 3 of diamonds is counted twice in this listing. Hence, there are 16 possibilities of drawing 3 or a diamond, so

P (3 or diamond) = 16÷ 52 = 4÷ 15 


Probability Rules
There are four basic probability rules. These rules are helpful in solving probability problems, in understanding the  nature of probability, and in deciding if your answers to the problems are correct.

Probability Rule 1:
The probability of any event E is a number (either a fraction or decimal) between and including 0  and 1.
This is denoted by 0≤P(E)≤1.

Rule 1 states that probabilities cannot be negative or greater than one.

Probability Rule 2:

If an event E cannot occur (i.e., the event contains no members in the sample space),
the probability is zero.

For Example:
When a single die is rolled, find the probability of getting a 9.

Solution

Since the sample space is 1, 2, 3, 4, 5, and 6,
it is impossible to get a 9.
Hence, the probability is P (9) = 0÷ 6

Probability Rule 3:

If an event E is certain, then the probability of E = 1.

In other words, if P (E) = 1, then the event E is certain to occur.
This rule is illustrated in the below Example

 Example: When a single die is rolled, what is the probability of getting a number less than 7?

Solution:

Since all outcomes, 1, 2, 3, 4, 5, and 6, are less than 7, the probability is

      P (number less than 7) = 6÷ 6 = 1

The event of getting a number less than 7 is certain.

Probability Rule 4:

The sum of the probabilities of the outcomes in the sample space is 1.

For example,
 In the roll of a fair die, each outcome in the sample space has a probability of 1÷ 6.
Hence, the sum of the probabilities of the outcomes is as shown.

Outcome	1	2	3	4	5	6	= 6÷6=1
Probability	1÷6	1÷6	1÷6	1÷6	1÷6	1÷6
Sum	1÷6    +   1÷6    +    1÷6         1÷6    +    1÷6    +   1÷6

Other important concepts will be followed tomorrow.

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