Concept of probability theory by dr shafia

probability

1. Probability Dr. Shafia Shaheen Associate Professor NIPSOM

2. Learning Objectives By the end of this class the students will be able to - understand the basic concept of random phenomenon and probability - understand different events of a random experiment - conceptualize types of probability - calculate different probabilities by addition and multiplication law

3. Probability theory is the branch of mathematics concerned with analysis of random phenomena . Random phenomena is a situation in which we know what outcome can occur, but we do not know which outcome will occur. The central objects of probability theory are random variables, random process and events .

4. Random experiment : An experiment is said to be an random experiment, if it’s outcomes cannot be predicted with certainty. Toss a coin twice Sample space : The set of all possible outcomes of an experiment is called the sample space. S = [HH, HT, TH, TT] Event : Every subset of a sample space is an event. A = (HH),

5. Venn Diagram A  Venn diagram  is a diagram that helps us visualize the logical relationship between  sets  and their elements. A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U. All the other sets are represents as circles within the universal set U.

6. Random variable : When the values of a variable obtained as a result of chance factors, so that they cannot be exactly predicted in advance, the variable is called random variable. 1. Discrete random variable 2. Continuous random variable

7. Definition Probability is the likelihood of occurrence of an event and is measured by the proportion of times it occurs. The probability P that an event E will occur, written P(E) is estimated by Probability =  

8. Two views of probability Subjective probability Based on personal judgements or experience Objective probability 1. Classical/Prior probability: In this type of probabilty any event has equal chance of occurring. Such as coin toss 2. Relative frequency/Posterior probability: Based on observations from probability experiments. Such as number of hypertensive patients among 100 respondents.

9. Properties of probability A probability value must lie between 0 and 1. A value 0 means the event can not occur. A value 1 means that the event definitely will occur. A value of .5 means that the probability that the event will occur is the same as the probability that it will not occur. The sum of the probabilities of all the events that can occur in the sample must be 1(or 100%).

10. Probability line is the chance that something will happen. It can be shown on a line.

11. Any planned activity or process that results in some definite outcome or event is called trial or experiment. Events posses certain characteristics 1. Mutually exclusive 2. Mutually non-exclusive 3. Exhaustive 4. Equally likely 5. Independent 6. Non – independent 7. Complementary events

12. Mutually exclusive events  are those events that do not occur at the same time. For example, when a coin is tossed then the result will be either head or tail, but we cannot get both the results .

13. Non-mutually exclusive events  are events that can happen at the same time. For example, a person may suffer from hypertension and diabetes at the same time. Non-mutually exclusive events  are events that can happen at the same time.

14. Equally likely event Equally likely  means that each outcome of an experiment occurs with equal probability. For example, if you toss a  fair , six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. 

15. Exhaustive events : All possible outcomes of an experiment constitute exhaustive events as one of them will definitely occur. Now, exhaustive events may or may not be equally likely events

16. Dependent and independent events Dependent event : When two events are dependent events, one event influences the probability of another event. A dependent event is an event that  relies on another event  to happen first.  Independent event : An independent event is an event that has  no connection  to another event’s chances of happening (or not happening). In other words, the event has no effect on the probability of another event occurring.

17. There are 15 marbles in a box, 6 marbles are blue and 9 marbles are yellow.

18. Complemenatry events : The sum of two probabilities is to be equal to 1. so, the probability of an event A is equal to 1 minus the probability of its complement, which is written P ( ) = 1 – P(A) P ( ) and P(A) are mutually exclusive.  

19. Types of probability Conditional probability : The size of group of interest may be reduced by conditions not applicable to the total group. When the probabilities are calculated with a subset of the total group as the denominator, the result is a conditional probability. The conditional probability is written as P(A│E) in which the vertical line is read ‘given’

20. Joint probability : Sometimes we want to find the probability that a subject picked at random from a group of subjects possesses two characteristics at the same time. Such a probability is referred to as a joint probability. The joint probability is written as P(EՈA). The symbol of joint probability is Ո, in which this symbol is read as either ‘intersection’ or ‘and’. The statement EՈA indicates the joint occurrence of conditions E and A. The numerator which satisfy both the conditions and the denominator will be the total outcomes.

21. Marginal probability : It is the probability in which the numerator of the probability is a marginal total from a table. P(D+) =   D+ D- Total T+ a b a+b T- c d c+d Total a+c b+d a+b+c+d

22. Rules of probability Addition rule Multiplication rule Family history of Mood disorders Early (E) Late (L) Total Negative (A) 28 35 63 Bipolar disorder (B) 19 38 57 Unipolar (C) 41 44 85 Unipolar bipolar (D) 53 60 113 Total 141 177 318

23. Addition rule This rule is applicable in both mutually exclusive and mutually non-exclusive events. This rule is characterized by presence of the term ‘or’ in between two events. We can state this probability in symbol as P(AUB), where the symbol U is read either as “union” or “or”.

24. Addition rule P(AUB) = P(A) + (B) P(AUB) = P(A) +P(B)- P(AՈB)

25. Example 1. We want to pick a person at random from the 318 persons. What is the probability that a person will be Early (E) onset or Late (L) onset? Solution: These two events are mutually exclusive So, P(E or L)/ P(EUL) = P(E) + P(L) = ( = .4434 + .5566 = 1  

26. Example 2. we want select a person at random from 318 persons. What is the probability that this person will be an Early (E) onset or will have no family history of mood disorder (A)? Solution: These two events are mutually non-exclusive. So, P(E or A)/ P(EUA) = P(E) + P(A) – P(EՈA) =( ) + ( = .4434+.1981-.0881 = .5534  

27. Multiplication rule This rule is applicable when two events out of two trials are considered. The events may be independent and dependent. This rule is characterized by the term ‘and’ in between the two events. For independent events P(A&B) or P(AՈB) = P(A).P(B) or P(B).P(A) For dependent events P(A&B) or P(AՈB) = P(A).P(B│A) or P(B).P(A│B)

28. Example 2. we want select a person at random from 318 persons. What is the probability that this person will be an Early (E) onset and will have no family history of mood disorder (A)? Solution: these are dependent events or outcomes. P(E & A) or P(EՈA) = P(E).P(A│E) = ( )x( =.4434x.1986 = .o881  

29. Exercise 1: Calculate P(D-), P(T+), P(T+ or D+), P(T- or D+), P(T+ and D+) Test result Truth Total D+ D- T+ 20 10 30 T- 15 105 120 Total 35 115 150

30. Exercise 2: In a study of violent victimization of women and men, their distribution is shown in the table No victimization Partners Nonpartners Multiple Total Women 611 34 16 18 697 Men 308 10 17 10 345 Total 919 44 33 28 1024

31. Suppose we a pick a subject at random from this group. What is the probability that this subject will be a women? What do we call the calculated probability? If we pick a subject at random, what is the probability that the subject will be a women and have experienced partner abuse? What do we call the calculated probability? Suppose we pick a man at random. What is the probability that he experienced abuse from nonpartners ? What do we call the calculated probability? Suppose we pick a subject at random. What is the probability that it is a man or someone who experienced abuse from a partner?

32. Best of Luck