Dr Hidayathulla Shaikh Probability

Objectives At the end of the lecture student should be able to Define probability Explain probability Enumerate laws of probability Discuss each laws of probability

Introduction Clinicians and researchers in biomedical and health sciences use their experimental data to draw conclusions and make inferences regarding the population they studied. Statistical techniques enable us to draw valid conclusions and make prediction on the characteristics of population. Proper understanding and correct application of statistical tests require knowledge of probability

Definition “Probability is defined as the relative frequency or probable chances of occurrence with which an event is expected to occur”. In everything we do from patient care to business we have chances of successful as well as failure events Eg dentist will be interested to know the chance that micro leakage will occur when occlusal surface is sealed with fissure sealant.

Probability (chance) explanation Probability is expressed by the symbol “p” and it ranges from 0 (zero) to 1 (one). When p = 0, it means there is no chance of an event happening or its occurrence is impossible. Eg – chances of survival after rabies is zero or nil. If p = 1, it means the chances of an event happening are 100% Eg death for any leaving being.

If probability of event happening in the sample is p and that of not happening is denoted by symbol q then q = 1 – p or p + q =1 One can estimate the probability by logic alone For eg probability p of drawing any one of 4 aces in one attempt from a pack of 52 cards is 4/52 = 1/13 And of not drawing is q = 1 – p = 1 – 1/13 = 12/13

Eg 2 chances of getting head or tail or getting male or female in one pregnancy is fifty – fifty or half that is p = ½ and q = ½. Where p is getting a male child and q is not getting a male child or getting a female child. Arithmetically we calculate the probability (p) or chances of occurrence of a positive event by the formula p = number of events occurring total number of trials

If a surgeon transplants kidney in 200 patients and gets success in 80 patients than probability of survival after operation is calculated by P = number of survival after operation total number of patients operated = 80/200 = 0.4 q = number of patients died total number of patients operated = 120/200 = 0.6

Laws of probability It is very important to have a clear concept of probability as it provides the basis for all the tests of significance. It is estimated by five laws of probability, normal curve and tables. The five laws of probability are 1) Addition law of probability 2) Multiplication law of probability 3) Binomial law of probability distribution 4) Probability from shapes of normal curve 5) Probability of calculated values from tables.

1) Addition law of probability If one event excludes the possibility of other event, then the events are called mutually exclusive. For eg getting head excludes the possibility of getting tail, birth of a male excludes birth of female, the blood group can be A, B, O or AB and so on….. Hence they all have equal chance of occurrence, if p1 is individual probability and p is total probability it is calculated as p = p1+p2+….Pn = 1, P = A+B+O+AB = 1 The word or is there when addition law is applied For eg a drug will cure or will have no effect on disease

2) Multiplication law of probability This law is applied to two or more events occurring together but they must be independent of each other, the word and is used in between the events. For eg 1 - A dice is thrown two times, what will be the probability of getting 5 and 3, So in first case probability of getting 5 in first throw is 1/6, and 3 in the second throw is 1/6, so probability of getting 5 in first throw and 3 in second throw is 1/6 x 1/6 = 1/36. Eg2 - gender and blood group are independent events, so what will be the probability of child being boy with O blood group Probability of child being boy = ½, probability of O blood group ¼, So probability of boy with O blood group will be ½ x ¼ = 1/8.

3) Binomial law of probability distribution It is given by the formula (p+q) n where n is sample size or number of events such as births, tosses etc, p is probability of success, and q is probability of failure. Eg when two children are born one after the other, the possible sequence will be one of the following four – Chances of getting two male = ¼ = 0.25 (25%) Chances of getting two females = ¼ = 0.25 (25%) Chances of getting both gender = 1/4+1/4= 0.5 (50%)

4) Probability from shapes of normal curve If heights are normally distributed and total number of study people (200) are taken, then 50% people will be above mean height and 50% below the mean height. The range mean 1SD (Standard Deviation) covers 68% and mean 2SD covers 95% of people, so the probability of having height above mean 2 SD is 2.5% and height below mean 2SD = 2.5%

5) Probability of calculated values from tables Probability of the calculated values occurring by chance is determined by referring to the respective tables. Eg probability or chances of dying or survival at any age is determined from life table constructed on mortality rates of large population.