Probability: Introduction to Basic Concept

Uncertainty pervades all aspects of human endeavor. Probability is one of our most important conceptual tools because we use it to assess degrees of uncertainty and thereby to reduce risk. Whether or not one has had formal instruction in this topic, s/he is already familiar with the concept of probability since it pervades almost all aspects of our lives. With out consciously realizing it many of our decisions are based on probability. For example, when you study for an examination, you concentrate more on areas that you feel are likely to be covered on the test. You may cancel or postpone an out door activity if you believe the likelihood of rain is high.

In business, probability plays a key role in decision-making. The owner of a retail shoe store, for example, orders heavily in those sizes that s/he believes likely to sell fast. The owner of a movie theatre schedules matinees only during holiday seasons because the chances of filling the theatre are greater at that time. The two companies decide to merge when they believe the probability of success is greater for the consolidated company than for either independently.

Some important Definitions:

Experiment:
Experiment is an act that can be repeated under given conditions.
Usually, the exact result of the experiment cannot be predicted with certainly.
Unit experiment is known as trial. This means that trial is a special case of experiment. Experiment may be a trial or two or more trials.

Outcomes:
The results of an experiment are known as outcomes.

Sample space & Sample point:
A sample space of an experiment is a set or collection of all possible outcomes of the same experiment and is usually denoted by the symbol .
Each outcome of an experiment can be thought of as a sample point or element in the sample space.

Events:
One or more outcomes of an experiment constitute an event.
Events re generally denoted by capital letter A, B, C, etc.

There are two types of events
Simple event: an event is called simple event if it contains only one sample point.
Compound event: An event is called compound event if it contains more than one sample point.

Mutually exclusive event or outcome:
Two outcomes or events are said to be mutually exclusive event or outcome if they cannot turn up at the same time.

Equally likely event or outcome:
Two or more events or outcomes are said to be equally likely if they have the same probability of occurrence.

Example: i. Tossing of a coin is a trial and getting head and tail are outcomes. ii. Throwing a die is a trial and obtaining 1 or 2 or 3 or 4 or 5 or 6 is an out come. iii. Drawing three balls from a bag containing 3 red and 4 black balls is a trial and getting 1 red and 2 black balls is an outcome. Different approaches in defining probability:
There are different approaches in defining probability are: 1. Classical or Mathematical or a priori probability 2. Empirical or frequency probability 3. Subjective probability

1. Classical or Mathematical or a priori probability:

If there are n mutually exclusive, equally likely and exhaustive outcomes of an experiment and if m of this outcomes are favorable to an event A, then the probability of the event A which is denoted by P(A) is defined by

| Favorable outcomes of an event A Total number of outcomes of the experiment |

There are three drawbacks of classical definition of probability: i. The classical probability fails to define probability when the total number of possible outcomes is infinite. ii. The classical definition leaves us completely helpless when the possible outcomes are not equally likely. iii. It is not always possible to enumerate all the equally likely cases. 2. Empirical or frequency probability: If an experiment is repeated a large number of times (n times) under the same conditions and an event A occurs m times then according to empirical or frequency approach of probability the probability of the event A | Number of times event A occurs. Total number of trials. | Drawbacks of the empirical probability: i. In practice, it is not possible to repeat the experiment an infinite number of times under the same conditions to get the probability. ii. It is not clear how large n should be before we are certain that the probability, p is close to the limiting of as . 2. Subjective probability: The probability that a person assigns to an event on the basis of his own judgment, beliefs and information about the event is known as subjective probability. The subjective probability has the following drawbacks, i. It varies from individual to individual as it depends on individual’s judgment and belief, ii. It has no objective basis. Sampling with replacement: If the elements of a sample are drawn randomly one by one and after each draw the element is returned to the population then the drawing is said to be done with replacement and the process of having the sample is called random sampling with replacement. Sampling without replacement: If the elements of a sample are drawn randomly one by one and after each draw the element is not returned to the population then the drawing is said to be done with out replacement and the process of having the sample is called random sampling with replacement. Example: A bag contains 4 white and 6 black balls. If one ball is drawn at random from the bag, what is the probability that it is i. Black, ii. White, iii. White or black and iv. Red. Answer: i. Let A be the event that the ball is black, then the number of outcomes favorable to A is 6. Hence | Favorable outcomes of an event A = Number of black balls Total number of outcomes of the experiment = Total number of balls | ii. Let B be the event that the ball is white, and then the number of outcomes favorable to B is 4. Hence | Favorable outcomes of an event B = Number of white balls Total number of outcomes of the experiment = Total number of balls | iii. Let C be the event that the ball is black or white and then the number of outcomes favorable to C is 10. Hence | Favorable outcomes of an event C = Number of white or white balls Total number of outcomes of the experiment = Total number of balls | iv. Let D be the event that the ball is red, and then the number of outcomes favorable to B is 4. Hence | Favorable outcomes of an event D = Number of red balls Total number of outcomes of the experiment = Total number of balls |

Example: A card is drawn from a pack of 52 cards. Find the probability that it is i. A red card, ii. A spade, iii. An ace, iv. Not a spade and v. a king or a queen. Answer: When a card is drawn from a pack of 52 cards, the total number of equally likely and mutually exclusive outcomes are 52. That is here i. Let A be the event drawing a red card. There are 26 black card and 26 red cards in a pack and any one of the red cards can be drawn in 26 ways. Hence

| Favorable outcomes of an event A = Number of red cards balls Total number of outcomes of the experiment = Total number of cards |

ii. Let B be the event drawing a spade. There are 13 spades. Hence

| Favorable outcomes of an event B = Number of spades Total number of outcomes of the experiment = Total number of cards |

iii. Let C be the event drawing an ace. There are 4 spades. Hence

| Favorable outcomes of an event C = Number of ace Total number of outcomes of the experiment = Total number of cards |

iv. Let D be the event drawing a card that is not a spade. There are 39 cards that is not spade. Hence | Favorable outcomes of an event D = Number of cards not spade Total number of outcomes of the experiment = Total number of cards |

v. Let E be the event drawing a card will be either king or queen. There are 4 kings and 4 queens. Hence | Favorable outcomes of an event E = Number of kings and queens Total number of outcomes of the experiment = Total number of cards |

Example: two balanced dice, one black and one red are thrown and the number of dots on their upper faces are noted, let b be the outcomes of the black die and r be the out comes of the red die. Now answer the following: i. List a sample space of the experiment. ii. What is the probability of throwing a double? iii. What is the probability that the sum is 5, that is ? iv. What is the probability that the sum is even? v. What is the probability that or ? vi. What is the probability that the number on the red die is at least 4 greater than the number on the black die.

Answer: i. If two balanced dice, one black and one red are thrown and the number of dots on their upper faces are noted, also if b be the outcomes of the black die and r be the out comes of the red die. Then the sample space for the given experiment will be as follows: | 1 | 2 | 3 | 4 | 5 | 6 | | | | | | | | 1 | 1,1 | 1,2 | 1,3 | | | 1,6 | 2 | 2,1 | | | | | | 3 | | | | | | | 4 | | | | | | | 5 | | | | | | | 6 | 6,1 | 6,2 | | | | 6,6 |

ii. Let the event A = {the two dice shows the same number}
= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}; There fore iii. Let the event B = {The sum of the two dies is 5, that is the two dice shows the same number}
= {(1,4), (2,3), (3,2), (4,1)}
There fore iv. Let the event C = {The sum of the two dies is even}
= {(1,1), (1,3), (1,5), (2,2), (2,4), …, … , … ,(6,4), (6,6), }
There fore

v. Let the event D = {(b,r)| or }
= {(1,1), (1,2), (1,3), (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), (5,1), (5,2), (6,1), (6,2)}
There fore

vi. Let the event E = {} = {(1,5), (1,6), (2,6)}
There fore

Rules of probability: 1. Addition rule: a. General Case (Events that are not mutually exclusive):
With in the sample space for any experiment, we may define any number of events; these events may or may not be mutually exclusive. The general addition rule for the events that are not mutually exclusive is given as follows:

b. Special case (Events that are mutually exclusive):
For mutually exclusive events the probability of either event occurring is simply the sum of their separate probability, as given below:

2. Multiplicative rule: a. Independence of events:
If the two events are independent then the probability that both the event occurs is given by

b. Events that are not independent:
If A and B are two not independent events then the probability that A occurs given that B has already occurred is called the conditional probability of A given B and is symbolically denoted by

Similarly the conditional probability of B given A and is symbolically denoted by

Some basic concepts of set theory: Set: A set is a well-defined collection of objects, which have some properties in common. Example: a) The books in a library. b) Set of all natural numbers. c) The set of vowel in English alphabet: a,e,,o and u . Universal Set: A set is called universal set if it contains all possible objects under consideration in a given discussion. Universal set is generally denoted by or or . If we through a die and observe the numbers appears on top then the universal set will be . Subset: A set B is said to be a subset of C if every elements of B is also an element of C. symbolically, we write this . Example: Let and , then B is a subset of set A. Venn Diagram: The diagram by which we can show the relationships between subsets and the corresponding universal set is called Venn - diagram. Usually, the universal set is represented by a rectangular and the subsets of it are represented by circle drawn inside it. Some Basic Set operations: Complementary set: If A is a subset of the universal set , then the complement of A with respect to is the set of all elements that are in but not in A. Let and , then the complement of the set is denoted by or is . Union of sets: If A and B be two subsets of , then the union of A and B is a set which contains all the elements in A and B or both. Let and , . Then the union of the two sets denoted by . Intersection of sets: If A and B be two subsets of , then the intersection of A and B is a set which contains all the elements that exists both in A and B. Let and , . Then the union of the two sets denoted by . Disjoint or mutually exclusive sets: If A and B be two subsets of , then A and B are said to be disjoint or mutually exclusive sets if they have no elements in common. That is, if . Let and , . Then the union of the two sets denoted by .

Assignment on Probability

1. Three thousand tosses of a certain coin gave 1800 heads. What is the probability of head? Is it classical or empirical or subjective probability?

2. A box contains 4 red balls, 6 black balls and 2 white balls. What is the probability of drawing a red ball? Is it classical probability?

3. The director of a nuclear plant feels that the probability of completing the new plant is . What kind of probability is this?

4. A card is drawn randomly from a bridge of deck. What is the probability that it will be i. Queen, ii. Queen of heart, iii. Heart and iv. Heart or spade?

5. A palmist told Fahim of BRAC University that he has a 10 to 3 chance of getting marry this year. What is the probability that PEU will not get married this year?

6. A balanced die is thrown. What is the probability of getting even number of points on the face? ? Is it classical or empirical or subjective probability?

7. A fair coin is tossed three times. Construct the sample space of the experiment. Hence find: i. A = {At least one tail}, B = {At most three heads}, C = {One head and two tail}, D = {One tail and two heads} and E = {Shows the same face}. 8. For problem 7 find the probability corresponding to the event and . 9. Two balanced dice are thrown and the number of dots on their upper faces is noted. Write down the sample space for the experiment and determine the probability of the following: i. First die shows odd number and second die show even number. ii. Outcomes of the first die are less than that of second die. iii. Find the probability that the sum of the outcome is even number. iv. Find the probability that the sum of the two outcomes is greater than 10. 10. Roll two dice and the number of dots on their upper faces is noted. Write down the sample space for the experiment and determine the probability of the following events: v. Event A = {Second die shows a number that is not divisible by 3}. vi. Event B = {Outcome of the first die is divisible by 3}. vii. Find viii. If the sum of the two values is 8, find the probability that one of the values is 3. ix. Sum of the points on the two dice is 10 or greater if a 5 appears on the first die.

11. An urn / box contains 7 red and 3 black marbles. Three marbles are drawn from the box one after another. Find the probability that the first two are red and the third is black. 12. An urn / box contains 75 marbles, 35 of them are blue and 25 of these blue marbles are swirled. The rest of them are red and 30 of the red ones are swirled. The marbles that are not swirled are clear. What is the probability of drawing: i. A blue marbles from urn ii. A clear marble from the urn iii. A blue, swirled marble iv. A red clear marble v. A swirled marble 13. If from a standard deck of cards a card is drawn then determine the probability of the following events: i. A seven ii. A black card iii. An ace or a king iv. A black 2 or a black 3 v. A red face card (king, queen or jack)