A powerful powerpoint presentation on probability
- 1. By: Manas Kumar Das MPharm '/Sf Year (Pharmaco/Ofl» Regd No. : 1661611@01 M .Pharm 1st Semester Seminar Subf ed :Biostatistics Top;c :ProbabU jty
- 2. Probability Probability is the likelihood of something happening in the future. It is expressed as a number between zero (can never happen) to 1 (will always happen). It can be expressed as a fraction, a decimal, a percent, or as "odds".
- 3. Probabi l i ty Li ne: Sometimes you can measure a probabil ty with a numberlike "10% chance rain", or you can use words such as impossible, unlikely, possible, even chance, likely and certain. Examp le: "It is unlikely to rain tomorrow". 0 Impos sible. U nlikely t t t Even Cho.nee Ce-rto1n 0 - - l Probability is always between 0 and 1
- 4. Some common terms related to probability : Experi ment: s a situation involving chance or probabil ity that leads to results called outcomes Outcome: A possible result of a random exper iment. Equally likel y outcomes: All outcomes w it h equal probability.
- 5. Some common terms related to probability (contd .) .,. Sampl e space: The possible set of outcomes of an experiment is known as sample space. . Example: choosing a card from a deck .,. There are 52 cards in a deck (not including Jokers) ,... So the Samp le Space is all 52 possible cards: {Ace of Hearts) 2 of Hearts, etc ... J.
- 6. ..,.Event: One or more outcomes in an experiment. ..,.Sample point: Each element of the samp le space is called a sample point. Ex: 1 1 > the 5 of Clubs is a sample point 1 1 > the King of Hearts is a sample point.
- 7. Ty pes of Proba bi li ty :Three types of probability are t here l l J i i > Classical definit ion of probability l l J i i > Statistical or Empir ical definition of probabi ity l l J i i > Subjective probabi ity
- 8. Classica l Proba bili ty: .. No of favourable outcomes Total no of outcomes P(E) = Ex: If we wanted to determine the probability of getting an even number when rolling a die, 3 would be the number of favorable outcomes because there are 3 even numbers on a die (and obviously 3 odd numbers). The number of possible outcomes would be 6 because there are 6 numbers on a die. Therefore, the probability of getting an even number when rolling a die is 3/6, or 1/2 when you simplify it.
- 9. E MPIRICAL P ROBABILITY The second one is empirical probability that is based o past experience. The empirical probability, also kno as relative frequency, or experimental probability P (E ) = # of times event E occurs total # of observed occurrences For example: (1) 383 of 751 business graduates were employed in the past. The probability that a particular graduate will be employed in his or her major area is 383/751 = 0.51 or 51°/ o. (2) The probability that your income tax return will be audited if lhere are two million mailed to your district office and 2,400 are to be audited is 2,400/2,000,000 = 0.0012 or 0.12°/o. n w n .
- 10. Subjective Probability • Subjective Probability is coming from person's jt1dgment or experience. • Example: - Probability of landing on "head" >vhen tossing a COlD . - Probabili ty of winning a lottery. - Chance that the stock n1arket goes down in com10g year.
- 12. Independent events: .,. If two events, A and 8 are independent then the joint probabi lity is ... P(A or B)=P(An B)=P(A)P(B) .,. for example, if two coins are flipped the chance of both being heads is ...L x _!_ = , - 2 2 4
- 13. Mutuall y excl usi ve events: .,. If either event A or event B occurs on a single performance of an experiment this is called the union of the events A and B denoted as P(A U B). If two events are mutually exclusive then the probability of either occurring is I" P(A or B)=P(A U B)=P(A)+P(B) • For example, the chance of rol ling a 1 or 2 on a six - sided die is .,_ P(1 or 2) = P(1) + P(2) = 1 /6 + 1/ 6 = 1 / 3
- 14. Not mutually exclusive events : • If the events are not mutually exclusive then • P(A or B) = P(A) + P(B) - P(A) and P(B) • Example - when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K ) (or one that is both) is 13/ 52 + 12/ 52 - 3/ 52 = 11/ 26 • because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but shou d only be counted once
- 15. Condi tional Probabi l ity: "' Two events A and B are said to be dependent when B can occur only when A is known to have occurred (or vice versa) .The probability attached to such t hat event is called conditional probabi lity and denoted by P(A/B). "' If two events A and B are dependent then the conditional probability of B given A is P(AB) - P(A) P(B/ A) -
- 16. • EX: A bag contain 5 white ba lls and 3 black balls. Two balls are drawn at random one after the other withou t replacement . Find the probabili ty that both ball drawn are black. • Sol : Probability of drawing a black ball in the first attempt is P(A) =3/ 5+3 =318 . Probability of drawing the second black ball given that f irst ball drawn is black P(B/A) = 2/ 5+2 = 2/ 7 The probability that both balls drawn are black is given by P(AB) = P(A) x P( B/A) = 3/8 x 2/ 7 = 3/28
- 17. Bayes' Theorem : L ikelihood Probability of cotlecung lhas data When our hypothe slSIs true P(H iD) - P(DIH) P(H) P(D) li!!:C!OC The probabl ltty of ou r hypothes s being true g iv e n the data collected n..o l t N Pdoc The p robability o r the hypothesis be ing true before collecllng data M rgjnS!J What ls the ptobab<bty o r col lectin g this d ata under all poss b le hypotheses?
- 18. Bayes Theorem Example S dc n d a « W ' I the- cwa, our priof btticf1 Q ,I bo I n . prk:lr JJ(CbllbilitydbttilluUCln Uat t Cf'lu w we e aboJt ll'r l'llinown fcnt.u'n. N t t t lNdat.a reoviW!d M1 s roc:tpt I C id by • OuU on O¥ef CM r i"""""' f P(C)P(X IC) P(C I X ) = P(X ) I post u ior u p rior > < likel ihood I Given: ¥' A doctor knows that meningitis causes stiff neck So: of the time - 11.,. ,,,..., "' Prior probability or any patient having meningitis Is 1/50,000- "' Prior probability or any patient having stiff neck Is 1120 " ''" ' + - - " '" " If a patient has stiff neck, what 's the probabili ty he/she has meningitis ? P(M I S ) = P(S I M )P (M ) = 0.5x I/ 50000 = O.OOO P(S) 1/20
- 19. Application of Proba bi li ty: " Applications of probability in analysis. " Point processes, random sets, and other spatial models. " -Branching processes and other models of population growth. 1 > -Genetics and other stochastic models In biology. 1 > -Information theory and signal processing • -Communication networks " -Stochastic models In operations research.
- 20. Reference: I> S C Gupta; Statistical Method; Sultan Chand & Sons Educational Publishers New Delhi; 2010; P.753·803 1 > P N Arora, S Arora, S Arora; Comprehensive Statistica l Method; S Chand & Company LTD; 2012; P . 11.3·11.101 1 > Journal of Probability and Statistics,8th Ed ition. Page 26-27. I> William Feller, ..An I ntroduction to Probability Theory and I ts Applications", (Vol 1), 3rd Ed, (1968)
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