Basic probability Concepts and its application By Khubaib Raza
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The document provides an overview of probability, detailing its historical development, definitions, properties, and real-life applications, particularly in computer science. It explains key concepts such as sample space, events, and the differences between probability and permutation, supported by various examples and exercises. Additionally, it highlights the significance of probability theory in fields like machine learning, information retrieval, and computer vision.
Basic probability Concepts and its application By Khubaib Raza
- 2. Meet Our Team Student of Government College University BSCS E2 Yaseen Saleem Ahtsham Naseer Roll-no 280 Roll-no 262 Roll-no 256 Roll-no 210 Roll-n0 232 Suleman Khan Ali Ahsan Khubaib Raza
- 3. Today Learn Attention Must require Introduction Probability and its Examples Probability and its Properties Difference between Probability and Permutation Application in Computer Science
- 4. INTRODUCTION The theory of probability was first developed in the seventeenth century when certain gambling games were analyzed by the French mathematician Blaise Pascal. It was in these studies that Pascal discovered various properties of Probability. In the eighteenth century, the French mathematician Laplace, who also studied gambling, gave definition of the probability as the number of successful outcomes divided by the number of total outcomes. (Real life Example).
- 5. Sample space and Event An experiment is a procedure that yields a given set of possible outcomes. The sample space of the experiment is the set of possible outcomes An event is a subset of the sample space. Before we start Probability understand this Concpet
- 6. Event and Sample Spaces Sample Space The sample space is the set of all possible outcomes. Simple Events The individual outcomes are called simple events. Event An event is any collection of one or more simple events
- 7. EXAMPLE Solution: P(heads) = 1/2 P(tails) = 1/2 Property: If you add these two up, you will get 1, which means the answers are probably right. If I flip a coin, what is the probability I get heads? What is the probability I get tails? Remember, to think of how many possibilities there are.
- 8. EXAMPLE When a die is tossed the sample space S of the experiment have the following six outcomes. S = {1, 2, 3, 4, 5, 6} Let E1 be the event that an even number occurs, E2 be the event that an odd number occurs, Then, E1 = {2, 4, 6} E2 = {1, 3, 5} What are the total possible outcomes if a dice is rolled?
- 9. Probability The possibility of an event to occur in a sample space is known as Probability. Let S be a finite sample space such that all the outcomes are equally likely to occur. EXAMPLE: What is the probability of getting a number greater than 4 when a dice is tossed? SOLUTION: When a dice is rolled its sample space is S={1,2,3,4,5,6} Let E be the event that a number greater than 4 occurs. Then, E = {5, 6} Hence, possibility that number will be greater than 4 is, Probability=1/3
- 10. Probability Properties of Probability • Probability is never negative • Probability is never more than one. • Probability always lies between 0 and 1 • Probability of impossible event is always equals to 0 • Probability of sure event is always equals to 1 • Sum of probability is always equal to 1 (p=1). • Let p=Probability of interested events q=Probability of uninterested events Then, p+q=1
- 11. EXERCISE SOLUTION: i) There are four Ace’s in a deck of cards(as ace’s are 4 in 52 cards); ii) There are two red Kings in a deck of cards(as two red King cards are there); One card is drawn at random from an ordinary deck of 52 cards. Find the probability p that (i) It is Ace (ii) It is Red King 13 1 52 4 drawnbecancardswaysofnumberTotal drawnbecansace'waysofNumber p 26 1 52 2 drawnbecancardswaysofnumberTotal drawnbecanKingsRedwaysofNumber p
- 12. EXAMPLE Solution: • Since there are four red balls and five blue balls so if we take out one ball from the box then there is possibility that it may be one of from four red and one of from five blue balls hence there are total of nine possibilities. Thus we have • The total number of possible outcomes = 4 + 5 = 9 • Now our favorable event is that we get the blue ball when we choose a ball from the box. So we have • The total number of favorable outcomes = 5 • Now we have Favorable outcomes 5 and our sample space has total outcomes 9 .Thus we have • The probability that a ball chosen = 5/9 An box contains four red and five blue balls. What is the probability that a ball chosen from the box is blue?
- 13. EXAMPLE SOLUTION: The possible outcomes of this experiment are red, green, blue and yellow. Total = 6+5+8+3 = 22 P(Red) = No. of ways to chose red/Total Marbles = 6/22 = 3/11 P(Green) = No. of ways to chose green/Total Marbles = 5/22 P(Blue) = No. of ways to chose blue/Total Marbles = 8/22 = 4/11 P(Yellow) = No. of ways to chose yellow/Total Marbles = 3/22 A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?
- 14. Example Choose a number at random from 1 to 5. What is the probability of each outcome? What is the probability that the number chosen is even? What is the probability that the number chosen is odd? SOLUTION: The possible outcomes of this experiment are 1, 2, 3, 4 and 5. Here, P(1)=P(2)=P(3)=P(4)=P(5) = Probability for any Number/Total Numbers = 1/5 P(Even) = No. of Even Numbers/Total Numbers = 2/5 P(Odd) = No. of Odd Numbers/Total Numbers = 3/5 Hence, the outcomes 1, 2, 3, 4 and 5 are equally likely to occur as a result of this experiment. However, the events even and odd are not equally likely to occur, since there are 3 odd numbers and only 2 even numbers from 1 to 5
- 15. Comparison Probabilty and Permutation Defination of Probability Formula of probability Example A fair coin is tossed find probability of (1) Head appears (2) Tail appears (3) Head or Tail appears Order Here order does’nt matter. Defination of permutation Formula of Permutation P(n,r) = n(n-1)(n-2)….(n-(n-r)) Example How many possible permutations can be made by three letters . A B C ? Here n=3 Order Here strictly we can say that order matters alot )( )( inoutcomestotalofnumbrthe inoutcomesofnumberthe )( Sn En S E EP
- 16. Comparison Probabilty and Permutation Keep in mind Tree diagrams in probability Answer judgment The answer of sum of probabilities of an event are equal to 1. Answer Range Here the answer of probability lies between 0 and 1. Tree Diagrams Here we can’t easily explain our answer with tree diagrams Answer Judgment Here we can’t judge the answer before solving Answer Range The answer of permutation lies in between the set of Natural numbers N
- 17. Probability in Computer Science • During the past two decades probability theory has come to play an increasingly important role in many areas of Computer Field. • Machine Learning(Data mining ) • Information Retrieval (Web ..) • Computer Vision • Robotics • Image Recovery • Classification and clustering
- 18. Probability in Computer Science Examples: Transfer data over the net
- 19. Probability in Computer Science Examples: Error Correction in Hard disks Google algorithms
- 20. Thank You WE ARE FIVE K hubaib A li S uleman A htesham Y aseen WeareFive@kasay.com We are Five We are Five