1 Probability Distributions GTECH 201 Lecture 14

2 Probability Rules P <= 0 <= 1 P (E) = Frequency of that one outcome Number of possible outcomes for mutually exclusive, complementary events P (A) + P (Ā) = 1 P (Ā) = 1 – P (A) P (A or B or C or….) = P (A) + P (B) + P (C) +.. P (A and B and C and …) = P (A) · P (B) · P (C) · … where A, B, C are independent

3 When Events are NOT Independent P (A) or P (B) = P (A) + P (B) – P (A & B) We are selecting cards from a set of 52 playing cards. What is the probability that the card selected is either a spade or a face card P (spade) = P (face card) = P (spade and face card) = Therefore P (spade) or p (face card) =

4 Another Example 1990 arrest data shows that: 79.6 % of people arrested were male 18.3 % were under 18 years 13.5 % were males under 18 years If you select an inmate at random, what is the probability that the person is either male OR under 18? P (male or under 18) = P (male) + P (under 18) – P (under 18) P (male) = P (less than 18) = P (male and less than 18) = → P (male or less than 18) = 0.844

5 Probability Distributions Over time, with enough data We recognize patterns Probability of outcomes are consistent E.g., tossing a coin, H or T Patterns are probability distributions Familiar with the bell curve For discrete outcomes Discrete probability distribution Continuous outcomes Continuous probability distribution

6 Binomial Distribution Discrete probability distribution Events have only 2 possible outcomes binary, yes-no, presence-absence Computing probability of multiple event P (x ) = n ! p x q n -x x !(n –x )! where: n = number of events or trials p = probability of the given (successful) outcome in a single trial q = p bar = (1-p) x = number of times the given outcome occurs within n trials n! = n factorial

7 Factorial n factorial is written as n! Definition: The factorial of a natural number n is the product of all non-zero numbers less than or equal n: When n = 0, 0! = 1 Therefore, when n = 1, n! = 1x(0)! or n! = 1x(1) or 1! = 1 When n = 2, 2! = 2x(1!) = [2x(1)], 2! = 2 When n = 3, 3! = 3x(2!) = 3x(2)x(1), 3! = 6 For n > 0, n! = n (n -1)! = [n x (n -1) x (n -2)………..x (2) x (1)] Following this, when n = 5, 5! = ?

8 The Poisson Distribution Used to analyze how frequently an outcome occurs during a certain specified time period, or across a particular area Understanding the probability of events that occur randomly over time or space Revisit this distribution during the sessions on spatial statistics

9 Binomial Probability Assumptions n identical trials are to be performed Two outcomes, success or failure are possible for each trial The trials are independent The success probability p, remains the same from trial to trial

10 To Find Binomial Probability Step 1Identify a success Step 2Determine p, the success probability Step 3Determine n, the number of trials Step 4 Apply the binomial probability formula

11 Example The National Institute of Mental Health reports that there is a 20 % chance of an adult American suffering from a psychiatric disorder. Four randomly selected adult Americans are examined for psychiatric disorders. Find the probability that exactly three of the four people examined have a psychiatric disorder.

12 Following the Steps Outlined Step 1: Identify a success i.e., selected individual suffers a psychiatric disorder Step 2: Probability of success p = 0.2; Therefore q = = 0.8 (Here failure = selected individual does not suffer from a psychiatric disorder) Number of trials, n = 4 Now apply the formula

13 Binomial Probability P (X=3) = = = =2.56 % chance that exactly 3 people selected at random will suffer from a psychiatric disorder

14 See it Graphically S F

15 Calculate The Probability ssss = (0.2)(0.2)(0.2)(0.2) sssf = (0.2)(0.2)(0.2)(0.8) ssfs = ssff =

16 Sampling Population The entire group of objects about which information is sought Unit any individual member of the population Sample a part or a subset of the population used to gain information about the whole Sampling Frame The list of units from which the sample is chosen

17 Why do We Need Sampling ? Obvious reasons cost, practicality Accuracy Loss of the sample Issues related to undercounting Convenience sampling

18 Simple Random Sampling A simple random sample of size n is a sample of n units chosen in such a way that every collection of n units from a sampling frame has the same chance of being chosen

19 Random Number Tables A table of random digits is: A list of 10 digits 0 through 9 having the following properties The digit in any position in the list has the same chance of being any of of 0 through 9; The digits in different positions are independent, in that the value of one has no influence on the value of any other Any pair of digits has the same chance of being any of the 100 possible pairs, i.e., 00,01,02,..98, 99 Any triple of digits has the same chance of being any of the 1000 possible triples, i.e., 000, 001, 002, …998, 999

20 Using Random Number Tables A health inspector must select a SRS of size 5 from 100 containers of ice cream to check for E. coli contamination The task is to draw a set of units from the sampling frame Assign a number to each individual Label the containers 00, 01,02,…99 Enter table and read across any line , 48, 66, 94, 87, 60, 51, 30, 92, 97