1. Randomness and Probability
Lecture 6
Randomness and Probability

2. Random phenomena and probability
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Random phenomena and probability
With random phenomena, we can’t predict the individual outcomes, but we can hope to understand characteristics of their long-run behavior.
For any random phenomenon, each attempt, or trial, generates an outcome.
We use the more general term event to refer to outcomes or combinations of outcomes. 2

3. QTM1310/ Sharpe
Sample spaces
A sample space is a special event that is the collection of all possible outcomes.
We denote the sample space S or sometimes Ω (omega)
The probability of an event is its long-run relative frequency.
Independence means that the outcome of one trial doesn’t influence or change the outcome of another. 3

4. Law of large numbers The Law of Large Numbers (LLN) states that
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Law of large numbers
The Law of Large Numbers (LLN) states that
if the events are independent,
then as the number of trials increases,
the long-run relative frequency of an event gets closer and closer to a single (true) value.
Empirical probability is based on repeatedly observing the event’s outcome. 4

5. QTM1310/ Sharpe
Law of averages
Many people confuse the Law of Large Numbers with the so-called Law of Averages
The Law of Averages doesn’t exist.
Presumably, the Law of Averages would say that things have to even out in the short run.
No luck this time, more luck next time. 5

6. Types of Probability: theoretical probability
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Types of Probability: theoretical probability
The (theoretical) probability of event A occurring can be computed with the following equation:
𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 6

7. Types of Probability: personal probability
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Types of Probability: personal probability
A subjective, or personal probability expresses your uncertainty about the outcome.
Although personal probabilities may be based on experience, they are not based either on long-run relative frequencies or on equally likely events. 7

8. Probability rules: Rule 1
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Probability rules: Rule 1
If the probability of an event occurring is 0, the event can’t occur.
If the probability is 1, the event always occurs.
For any event A, also written as ∀𝑨
0≤𝑃(𝑨)≤1 8

9. Probability rules: Rule 2
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Probability rules: Rule 2
The Probability Assignment Rule
The probability of the set of all possible outcomes must be 1.
𝑃 𝑺 =𝑃(Ω)=1
where S represents the set of all possible outcomes and is called the sample space. 9

10. Probability rules: Rule 3
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Probability rules: Rule 3
The Complement Rule
The probability of an event occurring is 1 minus the probability that it doesn’t occur.
𝑃 𝑨 =1−𝑃( 𝑨 𝐶 )
where the set of outcomes that are not in event 𝑨 is called the “complement” of 𝑨, and is denoted 𝑨 𝐶 . 10

11. Probability Rules: Example
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Probability Rules: Example
Lee’s Lights sell lighting fixtures. Lee records the behavior of 1000 customers entering the store during one week. Of those, 300 make purchases. What is the probability that a customer doesn’t make a purchase?
If P(Purchase) = 0.30, then
P(no purchase) = 1 – P(Purchase) =1 – 0.30 = 0.70 11

12. Probability rules: Rule 4
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Probability rules: Rule 4
The Multiplication Rule
For two independent events A and B, the probability that both, A and B, occur is the product of the probabilities of the two events.
𝑃 𝑨 𝑎𝑛𝑑 𝑩 =𝑃 𝑨 ∗𝑃(𝑩)
provided that A and B are independent. 12

13. Probability Rules: Example
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Probability Rules: Example
Whether or not a caller qualifies for a platinum credit card is a random outcome. Suppose the probability of qualifying is What is the chance that the next two callers qualify?
Since the two different callers are independent, then
𝑃 𝑨 𝑎𝑛𝑑 𝑩 =𝑃 𝑨 ∗𝑃(𝑩)
P(customer 1 qualifies and customer 2 qualifies)
= P(customer 1 qualifies) x P(customer 2 qualifies)
= 0.35 x 0.35 = 13

14. Probability rules: Rule 5
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Probability rules: Rule 5
The Addition Rule
Two events are disjoint (or mutually exclusive) if they have no outcomes in common.
The Addition Rule allows us to add the probabilities of disjoint events to get the probability that either event occurs.
𝑃 𝑨 𝑜𝑟 𝑩 =𝑃 𝑨 +𝑃(𝑩)
where 𝑨 and 𝑩 are disjoint. 14

15. Probability Rules: Example
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Probability Rules: Example
Some customers prefer to see the merchandise but then make their purchase online. Lee determines that there’s an 8% chance of a customer making a purchase in this way. We know that about 30% of customers make purchases when they enter the store. What is the probability that a customer who enters the store makes no purchase at all?
P(purchase in the store or online)
= P (purchase in store) + P(purchase online) =
= 0.38
P(no purchase) = 1 – 0.38 = 0.62 15

16. Probability rules: Rule 6
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Probability rules: Rule 6
The General Addition Rule
The General Addition Rule calculates the probability that either of two events occurs. It does not require that the events be disjoint.
𝑃(𝑨 𝑜𝑟 𝑩)=𝑃(𝑨)+𝑃(𝑩)−𝑃(𝑨 𝑎𝑛𝑑 𝑩)
Recall
𝑃 𝑨 𝑎𝑛𝑑 𝑩 =𝑃 𝑨 ∗𝑃(𝑩) 16

17. Probability Rules: Example
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Probability Rules: Example
Lee notices that when two customers enter the store together, their behavior isn’t independent. In fact, there’s a 20% chance they’ll both make a purchase. When two customers enter the store together, what is the probability that at least one of them will make a purchase?
𝑃 𝑩𝒐𝒕𝒉 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆
=𝑃 𝑨 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒔 𝑜𝑟 𝑩 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒔
=𝑃(𝑨 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒔) +𝑃(𝑩 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒔) –𝑃(𝑨 𝑎𝑛𝑑 𝑩 𝒃𝒐𝒕𝒉 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆)
= – 0.20
= 0.40 17

18. Probability rules: EXAmple
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Probability rules: EXAmple
Car Inspections
You and a friend get your cars inspected. The event of your car’s passing inspection is independent of your friend’s car. If 75% of cars pass inspection what is the probability that
Your car passes inspection?
Your car doesn’t pass inspection?
Both cars pass inspection?
At least one of two cars passes?
Neither car passes inspection? 18

19. Probability rules: Example
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Probability rules: Example
You and a friend get your cars inspected. The event of your car’s passing inspection is independent of your friend’s car. If 75% of cars pass inspection what is the probability that
Your car passes inspection?
𝑃(𝑷𝒂𝒔𝒔) = 0.75
Your car doesn’t pass inspection?
𝑃(𝑷𝒂𝒔𝒔𝑪) = 1−0.75 = 0.25
Both cars pass inspection?
𝑃(𝑷𝒂𝒔𝒔)𝑃(𝑷𝒂𝒔𝒔) = (0.75)(0.75) =
At least one of two cars passes?
1 – (0.25)2 = OR
– =
Neither car passes inspection?
1 – = 19

20. QTM1310/ Sharpe
Contingency tables
Events may be placed in a contingency table such as the one in the example below.
As part of a Pick Your Prize Promotion, a store invited customers to choose which of three prizes they’d like to win. The responses could be placed in the following contingency table: 20

21. QTM1310/ Sharpe
Marginal probability
Marginal probability depends only on totals found in the margins of the table. 21

22. QTM1310/ Sharpe
Marginal probability
In the table below, the probability that a respondent chosen at random is a woman has a marginal probability of
𝑃(𝒘𝒐𝒎𝒂𝒏) = 251/478 = 22

23. QTM1310/ Sharpe
Joint probabilities
Joint probabilities give the probability of two events occurring together.
𝑃(𝒘𝒐𝒎𝒂𝒏 𝑎𝑛𝑑 𝒄𝒂𝒎𝒆𝒓𝒂) = 91/478 = 0.190 23

24. Conditional distribution
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Conditional distribution
Each row or column shows a conditional distribution given one event.
In the table above, the probability that a selected customer wants a bike, given that we have selected a woman is:
𝑃(𝒃𝒊𝒌𝒆|𝒘𝒐𝒎𝒂𝒏) = 30/251 = 0.120 24

25. Conditional probability
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Conditional probability
In general, when we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.”
𝑃 𝑩 𝑨 = 𝑃(𝑨 𝑎𝑛𝑑 𝑩) 𝑃(𝑨)
A probability that takes into account a given condition is called a conditional probability. 25

26. Probability rules: Rule 7
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Probability rules: Rule 7
The General Multiplication Rule
The General Multiplication Rule calculates the probability that both of two events occurs. It does not require that the events be independent.
𝑃 𝑨 𝑎𝑛𝑑 𝑩 =𝑃 𝑨 ∗𝑃(𝑩|𝑨) 26

27. Probability rules: Example
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Probability rules: Example
What is the probability that a randomly selected customer wants a bike if the customer selected is a woman?
𝑃 𝐵 𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴) =𝑃 𝑏𝑖𝑘𝑒 𝑤𝑜𝑚𝑎𝑛 = 𝑃(𝑏𝑖𝑘𝑒 𝑎𝑛𝑑 𝑤𝑜𝑚𝑎𝑛) 𝑃(𝑤𝑜𝑚𝑎𝑛) = 30/ /478 =0.12 27

28. Probability rules: Example
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Probability rules: Example
Are Prize preference and Sex independent? If so, P(bike|woman) will be the same as P(bike). Are they equal?
P(bike|woman)= 30/251 = 0.12
P(bike) = 90/478 = 0.265
0.12 ≠ 0.265
Since the two probabilites are not equal, Prize preference and Sex and not independent. 28

29. Independence Events A and B are independent whenever 𝑃(𝑩|𝑨) = 𝑃(𝑩)
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Independence
Events A and B are independent whenever
𝑃(𝑩|𝑨) = 𝑃(𝑩)
Independent vs. Disjoint
For all practical purposes, disjoint events cannot be independent.
Don’t make the mistake of treating disjoint events as if they were independent and applying the Multiplication Rule for independent events. 29

30. Constructing contingency tables
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Constructing contingency tables
If you’re given probabilities without a contingency table, you can often construct a simple table to correspond to the probabilities and use this table to find other probabilities. 30

31. Constructing contingency tables
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Constructing contingency tables
A survey classified homes into two price categories (Low and High). It also noted whether the houses had at least 2 bathrooms or not (True or False). 56% of the houses had at least 2 bathrooms, 62% of the houses were Low priced, and 22% of the houses were both. Translating the percentages to probabilities, we have: 31

32. Constructing contingency tables
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Constructing contingency tables
The 0.56 and 0.62 are marginal probabilities, so they go in the margins.
The 22% of houses that were both Low priced and had at least 2 bathrooms is a joint probability, so it belongs in the interior of the table. 32

33. Constructing contingency tables
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Constructing contingency tables
The 0.56 and 0.62 are marginal probabilities, so they go in the margins.
The 22% of houses that were both Low priced and had at least 2 bathrooms is a joint probability, so it belongs in the interior of the table.
Because the cells of the table show disjoint events, the probabilities always add to the marginal totals going across rows or down columns. 33

34. Constructing contingency tables: Example
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Constructing Contingency Tables
Constructing contingency tables: Example
A national survey indicated that 30% of adults conduct their banking online. It also found that 40% under the age of 50, and that 25% under the age of 50 and conduct their banking online.
What percentage of adults do not conduct their banking online?
What type of probability is the 25% mentioned above?
Construct a contingency table showing joint and marginal probabilities.
What is the probability that an individual who is under the age of 50 conducts banking online?
Are Banking online and Age independent? 34

35. Constructing contingency tables: Example
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Constructing Contingency Tables
Constructing contingency tables: Example
What percentage of adults do not conduct their banking online?
100% – 30% = 70%
What type of probability is the 25% mentioned above?
Marginal
Construct a contingency table showing joint and marginal probabilities. 35

36. Constructing contingency tables: Example
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Constructing contingency tables: Example
What is the probability that an individual who is under the age of 50 conducts banking online?
0.25/0.40 = 0.625
Are Banking online and Age independent?
No. P(banking online|under 50) = 0.625, which is not equal to P(banking online) = 0.30. 36

37. Quick summary Beware of probabilities that don’t add up to 1.
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Quick summary
Beware of probabilities that don’t add up to 1.
Don’t add probabilities of events if they’re not disjoint.
Don’t multiply probabilities of events if they’re not independent.
Don’t confuse disjoint and independent. 37

38. QTM1310/ Sharpe
Learning outcomes
Apply the facts about probability to determine whether an assignment of probabilities is legitimate.
Probability is long-run relative frequency.
Individual probabilities must be between 0 and 1.
The sum of probabilities assigned to all outcomes must be 1.
Understand the Law of Large Numbers and that the common understanding of the “Law of Averages” is false. 38

39. QTM1310/ Sharpe
Learning outcomes
Know the 7 rules of probability and how to apply them.
Know how to construct and read a contingency table.
Know how to define and use independence 39