PROBABILITY

Probability in Biostat

1. Probability Ms. Chaitali C. Dongaonkar

2. Introduction Measure of likelihood that an event will occur in random experiment Examples: Prediction about New drug, Estimation of production costs, forecasting vaccine failure Element of chance Ratio of number of favourable outcomes to the all possible outcomes i.e. favourable outcomes + unfavourable outcomes Let, “a” is number of favourable outcomes of an event “A” and “b” is number of unfavourable outcomes of an event “A” What will be the Probability of occurrence of A….???

3. Introduction Number of favourable outcomes/Total Number of outcomes a/a+b The Probability of unfavourable outcomes of A is given by, = Number of unfavourable outcomes/Total Number of outcomes = b/a+b Total Probability = a/a+b + b/a+b = 1 Theoretically, Total probability = 1 Practically, 0≤ P ≤ 1 Probability of event: ratio, fraction, percentage.  

4. Terminologies: Experiment or Trial: any procedure that can be infinitely repeated List of outcomes specified in advance Actual occurrence of outcome cannot be predicted in advance. Example: Probability of getting 5 on top of die. Sample Space: List of all possible outcomes of an experiments Example: if we throw die then sample space consists of { 1, 2, 3,4,5,6 } Event: The occurrence of particular outcome or combinations of outcomes Example: Tossing of coin, outcome of this will be “Head” or “Tail”

5. Example 1. In plastic container, there are 5 Crocin and 10 Aspirin Tablets. If one tablet is chosen at random, find the probability that Aspirin tablets or Crocin tablets are selected.

6. Conditional Probability Measure of probability of an event occurring Another event has already occurred

7. Probability Distribution Probability Distribution Discrete Probability Distribution 1. Binomial Distribution 2. Poisson Distribution Continuous Probability Distribution 1. Uniform Distribution 2. Exponential Distribution 3. Normal Distribution

8. Discrete Probability Distribution

9. Binomial Distribution Also known as Bernoulli's distribution Simply success or failure outcomes when experiments repeated several times Two possible outcomes, Prefix “bi” = Two/Twice Examples: Coin tossed two possible outcomes Head or Tail, Taking test two possible outcomes Pass or Fail, Lottery ticket Win money or Not Only success or failure represented by “Binomial Distribution” Success (p) and Failure (q), p + q = 1 (0.5 + 0.5 = 1) P (x success in n trials)= n C X P X q n-x n is number of trials, p is probability of success in trial, q is probability of failure in trial

10. Properties of Binomial Distribution Fixed number of observations or trials Each trial results as outcome as Success (s) or Failure (f) Each observation or trial is independent Probability of success exactly same from one trial to another i.e. DOES NOT CHANGE WITH TIME Mean = np, Variance = npq, Standard Deviation = where, n is number of trials, p is probability of success in trial, q is probability of failure in trial Mean shows average number of success, variance less than its mean p = q, symmetrical distribution, p < 0.5 = + ve skewness, p>0.5 = - ve skewness  

11. examples 1. The mean of binomial distribution is 40 and its standard deviation is 6. Calculate n, p, q.

12. examples 2. Multipunch tablet machine produces 12% defective tablets. What is the probability that out of random sample of 20 tablets produced by the machine, 4 are defective ?

14. Poisson Distribution Simeon Denis Poisson – in 1837, French mathematician For Number of discrete events in given period of time (fixed) Trials = large i.e. n is very large Success = very small i.e. 0 Used for rare events like number of people arriving in 1 hr, number of phone calls in a day, defective dosage per batch, number of microbes in samples Represents number of occurrence of an event in one unit of time

15. Poisson distribution

16. Poisson Distribution - Properties Events are discrete i.e. you can count them Events can not happen at the same time Events are independent Probability of two or more occurrences in a very small interval is close to Zero More symmetric as its mean or variance increases Uniparametric in nature Approximation for binomial distribution

17. example If the probability that an individual suffers a adverse reaction from a swine flu vaccine injection is 0.002, determine the probability that out of 1000 individuals (i) exactly two, (ii) at most two will suffer a adverse reaction.

18. Continuous RaNDOM VARIABLE

19. Normal Distribution Known as normal probability curve or Gaussian distribution Frequencies distributed evenly about mean of distribution Univariate distribution used for continuous random variables Real life situations like monthly salary of employees in a locality, marks of students in an entrance test Shape of normal distribution is bell shaped curve

20. Normal distribution - properties Mean = Mode = Median Curve is symmetric about the mean i.e. at the centre Exactly half values are to the left and half values are to the right The total area under curve is 1 or 100% Standard normal model = normal distribution with mean of 1 and standard deviation of 1 Curve: only one top point so its unimodal No skewness SD = determines width of the curve Fitting of actual observed frequency

21. The empirical rules tells what % of data that falls within a certain number of standard deviations from mean………

22. formula

23. example In male student population of 500, the mean height is 70.2 inches and variance is 10.5 inches. How many male students have height more than 73 inches? (Given: Area under normal curve corresponding to z = 0.8641 is 0.3039)