BASIC PROBABILITY Business Statistics
CONTENTS Data as a random sample Probability theory Be careful with probability Old exam question Further study
DATA AS A RANDOM SAMPLE In business and economics, the data represents a small sample of the phenomenon of interest poll with random 100 telephone calls ▪ ▪ interviews with random 25 customers batch of random 50 meals for food quality ▪ car accidents on 5 random days in a year ▪ etc. ▪ All are supposed to represent a bigger population the entire population of the US (or only those that have a telephone?) ▪ all of your customers (or all of your customers that have a phone, e- ▪ mail, or that visit your shop) all meals produced in some factory or some restaurant ▪ all accidents in a particular year, particular country, ... ▪ ▪ etc.
DATA AS A RANDOM SAMPLE Task: to infer properties of the population from the sample ▪ Inferential statistics ▪ example: 12 out of 100 random clients liked our new product – what proportion of the total population will like it? In order to get an understanding: study the variation of samples ▪ Probability theory ▪ example: if 15% of the population likes our new product, what is the probability that we find 12 in a random sample of 100?
PROBABILITY THEORY Sample space 𝑇 : all possible outcomes of a random experiment ▪ coin: 𝑇 = heads, tails ▪ die: 𝑇 = 1,2,3,4,5,6 or ▪ number of accidents on a day: 𝑇 ⊂ ℕ 0 ▪ body height: 𝑇 ⊂ ℝ + ▪ two dice: 𝑇 = or 1,1 , 1,2 , … , 6,6 ⊂ means “is a subset of”
PROBABILITY THEORY Probability 𝑄 : likelihood of a particular outcome coin: 𝑄 heads = 1 ▪ 2 die: 𝑄 3 = 1 ▪ 6 number of accidents on a day: 𝑄 1 = 0.19 (for example) ▪ body height: 𝑄 1.8304678 = 0 (why?) ▪ Types of probabilty ▪ a priori (classical theory) 𝑄 head = 1 2 or 𝑄 event = number of outcomes in event ▪ number of possible outcomes empirical ▪ 𝑄 event = number of occurrences of event ▪ number of observations subjective ▪ ▪ 𝑄 Italy will win next World Cup = ⋯
PROBABILITY THEORY Probability function 𝑄 event ▪ for every event 𝐵 ⊂ 𝑇 : 𝑄 𝐵 ≥ 0 ▪ for entire sample space 𝑇 : 𝑄 𝑇 = 1 ▪ for 𝐵 = ∅ : 𝑄 𝐵 = 0 ∅ is the empty set Events: ▪ elementary (e.g. for a die, outcome = 3) ▪ compound (e.g. for a die, outcome = even) Example: die with 𝑇 = 1,2,3,4,5,6 1 ▪ 𝑄 3 = 6 3 1 ▪ 𝑄 even = 𝑄 2,4,6 = 6 = 2 ▪ 𝑄 even or odd = 𝑄 𝑇 = 1 ▪ 𝑄 −1 = 𝑄 7 = 𝑄 2.43 = 𝑄 ∅ = 0
PROBABILITY THEORY The complement of an event 𝐵 is denoted by 𝐵 ′ and consists of everything in the sample space 𝑇 except event 𝐵 Since 𝐵 and 𝐵 ′ together comprise the entire sample space 𝑇 , 𝑄 𝐵 + 𝑄 𝐵 ′ = 1 or 𝑄 𝐵 ′ = 1 − 𝑄 𝐵
PROBABILITY THEORY The union of two events consists of all outcomes in the sample space 𝑇 that are contained either in compound event 𝐵 (e.g., “ ≤ 3 ”) or in compound event 𝐶 (e.g., “even”) or both ▪ denoted by 𝐵 ∪ 𝐶 ▪ pronounced “ 𝐵 or 𝐶 ” 4 1 “or” means “and/or”, so not 2 the exclusive or (as in “either 3 6 𝐵 or 𝐶 ”)
PROBABILITY THEORY The intersection of two events 𝐵 and 𝐶 is the event consisting of all outcomes in the sample space 𝑇 that are contained in both event 𝐵 and event 𝐶 denoted by 𝐵 ∩ 𝐶 ▪ ▪ pronounced “ 𝐵 and 𝐶 ” ▪ also known as the joint probability 4 1 2 3 6
EXERCISE 1 Given are two dies, their random outcomes are 𝑌 and 𝑍 . a. Find 𝑄 𝑌 = 2 ∩ 𝑍 = 3 ∪ 𝑌 = 3 ∩ 𝑍 = 2 b. Find 𝑄 𝑌 + 𝑍 = 5
PROBABILITY THEORY The general law of addition states that 𝑄 𝐵 ∪ 𝐶 = 𝑄 𝐵 + 𝑄 𝐶 − 𝑄 𝐵 ∩ 𝐶 ▪ when you add the 𝑄(𝐵) and 𝑄(𝐶) together, you count the 𝑄(𝐵 ∩ 𝐶) twice ▪ so, you have to subtract 𝑄(𝐵 ∩ 𝐶) to avoid over-stating the probability ▪ often, the right-hand side is easier to find
PROBABILITY THEORY Example: standard deck of cards ▪ 52 cards 4 ▪ 4 queens: 𝑄 𝑅 = 52 26 ▪ 26 red cards: 𝑄 𝑆 = 52 2 ▪ 2 red queens: 𝑄 𝑅 ∩ 𝑆 = 52 ▪ the probability that a random card is red or a queen: 4+26−2 28 𝑄 𝑅 ∪ 𝑆 = 𝑄 𝑅 + 𝑄 𝑆 − 𝑄 𝑅 ∩ 𝑆 = = 52 52 ▪ so 53.85%
PROBABILITY THEORY Events 𝐵 and 𝐶 are mutually exclusive (or disjoint) if their intersection is the null set ( ∅ ) that contains no elements ▪ if 𝐵 ∩ 𝐶 = ∅ , then 𝑄 𝐵 ∩ 𝐶 = 0 In the case of mutually exclusive events, the addition law reduces to: ▪ 𝑄 𝐵 ∪ 𝐶 = 𝑄 𝐵 + 𝑄 𝐶 Example: 𝑄 𝑅 ∪ 𝐿 = 𝑄 𝑅 + 𝑄 𝐿 because 𝑅 ∩ 𝐿 = ∅ ▪ more general: 𝑄 𝑅 ∪ 𝑅 ′ = 𝑄 𝑅 + 𝑄 𝑅 ′ because 𝑅 ∩ 𝑅 ′ = ∅ ▪
PROBABILITY THEORY Events are collectively exhaustive if their union is the entire sample space 𝑇 Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events ▪ for example, a car repair is either covered by the warranty ( 𝐵 ) or not ( 𝐵 ′ ) Mutually exclusive, collectively exhaustive events are a partition of the sample space
PROBABILITY THEORY The probability of event 𝐵 given that event 𝐶 has occurred ▪ written as 𝑄 𝐵 𝐶 ▪ pronounced as “probability of 𝐵 given 𝐶 ” ▪ for example, the probability of passing the statistics exam, given that you have passed the mathematics exam 𝑄 𝐵 𝐶 = 𝑄 𝐵 ∩ 𝐶 𝑄 𝐶 ▪ only defined when 𝑄 𝐶 > 0 ▪ this is a conditional probability
PROBABILITY THEORY Example: highs school drop-outs Facts about population aged 16-21 and not in college: ▪ unemployed (U): 12% ▪ high school drop-out (D): 30% ▪ unemployed high school drop-out: 4% What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout? 𝑄 𝑉∩𝐸 0.04 ▪ 𝑄 𝑉 𝐸 = = 0.30 = 0.133 𝑄 𝐸 Compare to 𝑄 𝑉 = 0.12 ▪ so, being a high school dropout is related to being unemployed Here is a first example of learning from data, although it is not yet inferential statistics (why not?)
EXERCISE 2 Given are two dies, their random outcomes are 𝑌 and 𝑍 . a. Find 𝑄 𝑌 + 𝑍 > 4 𝑌 < 2 b. Find 𝑄 𝑌 = 3 𝑍 ≤ 2
PROBABILITY THEORY Events 𝐵 and 𝐶 are independent if the 𝑄 𝐵 𝐶 = 𝑄 𝐵 (for 𝑄 𝐶 > 0 ) ▪ 𝑄 𝑉 𝐸 = 0.133 ≠ 𝑄 𝑉 = 0.12 ▪ so 𝑉 and 𝐸 are not independent ▪ that is, they are dependent Another way to check for independence: 𝑄 𝐵 ∩ 𝐶 = 𝑄 𝐵 𝑄 𝐶 ⇔ 𝑄 𝐵 ×𝑄 𝐶 𝑄 𝐵∩𝐶 = 𝑄 𝐵 𝐶 , ⇔ 𝑄 𝐵 = = 𝑄 𝐶 𝑄 𝐶 so 𝑄 𝐵 ∩ 𝐶 = 𝑄 𝐵 𝑄 𝐶 ⇔ 𝐵 and 𝐶 are independent even when 𝑄 𝐵 = 0 or 𝑄 𝐶 = 0
PROBABILITY THEORY Contingency tables (see also summarizing data) ▪ from frequencies ... ▪ to proportions
PROBABILITY THEORY Table contains ▪ joint probabilities, like 𝑄 Christ ∩ GATT = 0.36 ▪ marginal (simple) probabilities, like 𝑄 Christ = 0.49 Table can be used to find 0.36 ▪ conditional probabilities, like 𝑄 Christ GATT = 0.78 = 0.46 ▪ dependence: 𝑄 Christ GATT ≠ 𝑄 Christ ▪ later, we will develop a statistical test for assessing this for inference from the sample to the population
PROBABILITY THEORY Independence ▪ flipping a fair coin 4 times, what is the probability of obtaining 4 heads? ▪ you are given that the outcomes are independent ▪ 𝐼 1 means “ heads ” in experiment 1, 𝑈 1 “tails”, etc; ▪ 𝑄 4 heads = 𝑄 𝐼 1 𝐼 2 𝐼 3 𝐼 4 = 𝑄 𝐼 1 × 𝑄 𝐼 2 × 1 1 1 1 1 𝑄 𝐼 3 × 𝑄 𝐼 4 = 2 × 2 × 2 × 2 = 16 ≈ 6% Independence is often not realistic in business problems ▪ fashion: 𝑄 I buy a red shirt depends on what others do ▪ supply: 𝑄 I buy a red shirt depends on what is available in the shops ▪ combi: 𝑄 I buy a red shirt depends on if I also buy a red coat ▪ etc.
BE CAREFUL WITH PROBABILITY • Problem: aggregation Context: which of two medicines (A and B) is best? • Facts: • ▪ medicine A is better than B for treating cold ▪ medicine A is better than B for treating flue The most effective medicine ▪ but altogether B looks better Treatment Treatment A B Recovery Cold 93% 87% Flu 73% 69% 81 87 = 0.93 Both 78% 83% Treatment A Treatment B Yes No Tot Yes No Tot Cold 81 6 87 234 36 270 Flu 192 71 263 55 25 80 273 350 289 350
BE CAREFUL WITH PROBABILITY • Problem: aggregation Context: which of two football players (A and B) should • take a penalty? Facts: • ▪ in 2012 A scored 8/12=0.67 and B scored 3/4=0.75 ▪ in 2013 A scored 1/6=0.17 and B scored 1/5=0.20 ▪ so B was best in both years separately ▪ but in 2012-2013 A scored 9/18=0.50 and B scored 4/9=0.44 ▪ so A was best in the two combined years
BE CAREFUL WITH PROBABILITY • Problem: difference in point of view Context: what is the class size of an elementary school? • Facts: • ▪ there are 3 classes with 30 pupils and 3 classes with 10 pupils 3×30+3×10 ▪ the director tells that average class size is = 20 3+3 90×30+30×10 ▪ but pupils tell that average class size is = 25 90+30 What do we mean by “mean”? •
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