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Theory of Probability
Prof Tasneem Chherawala
NIBM
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Uncertain Events
• See a Zebra the classroom today?
• Win in a lottery?
• Toss a coin and get heads / roll a die (or two) and get 5/ pick a card from a deck and get an Ace?
• Make a 10% profit from investment in shares over 1 month?
• Face a default on money lent?
• Earn positive returns on a fixed deposit?
• Count the total number of hours today and get 24?
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Probability
• Probability quantifies how uncertain we are about a
future outcome / event
• A probability refers to the percentage chance that
something will happen, from 0 (it is impossible) to 1 (it is
certain to occur), and the scale going from less likely to
more likely
• An interpretation based on data - Probability can be
interpreted as the relative frequency of the outcomes
(values) of uncertain events (variable) after a great many
(infinitely many) repetitions / parallel independent trials of
an experiment
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Why Measure Uncertainty
• Is something at stake?
• To make tradeoffs among uncertain events
• Measure combined effect of several uncertain events
• To communicate about uncertainty
• To draw inferences about a population from a sample
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Probability Theory in Finance
• Financial decisions – a game of chance!
• What chance?– Probability of making or losing money in an investment!
• Why chance? – Uncertainty and variability of future events (price movements
and value of investments)
• Probability concepts help define financial risk by quantifying the prospects for unintended and negative outcomes (losses)
• Probability also quantifies expected values of future events, which gives us a fair estimate of current value of investment
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Probability Theory in Finance
• Probability theory applications:– Make logical and consistent investment decisions
– Manage expectations in an environment of risk
• Hypothesis: No system for making financial choices from those offered to us can both (1) be certain to avoid losses and (2) have a reasonable chance of making us rich
• Expected values in a probability model are the prices of alternative financial decisions
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Some Definitions
• Random Experiment: a process that leads to one of several possible outcomes
• Outcome: result of the experiment– Examples: Head in a coin toss, two heads when tossing two
coins, win in a lottery, 10% returns from the stock in one day, Borrower repays Re 1 from Rs. 100 borrowed at 12% rate of interest
• Sample Space: a set of all possible outcomes of the experiment – relates to population data– Examples:
• Toss one coin: S = {H, T}
• Toss two coins / toss a coin twice: S = {HH, HT, TH, TT}
• Lottery: S = {Win, Lose}
• One day stock returns: S = {-100% to +Infinity?}
• Repayment on Rs. 100 loan by borrower at 12% rate of int.: S = {0,1,2,….., 100, 101,…,112}
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Sample Space
• Finite sample space: finite number of outcomes in the space S.
• Countable infinite sample space: ex. natural numbers.
• Discrete sample space: if it has finite or countable infinite number of outcomes.
• Continuous sample space: If the outcomes constitute a continuum. Ex. All the points in a line.
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Events
• Event: subset of a sample space – a combination of possible outcomes of an uncertain process – Examples
• Head in one toss of a coin: E1 = {H}
• One head and one tail in two coin tosses: E2 = {HT, TH}
• Mean of the dice values greater than or equal to 5 in two dice rolls: E3 = {4.6, 5.5, 5.6, 6.4, 6.5, 6.6}
• One day stock returns greater than equal to 10%: E4 = {10% to +Infinity}
• Loss to bank if Borrower doesn’t repay principal: D = {1%,2%,…..,99%,100%}
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Events
• Mutually Exclusive Events: events which have no outcome in common and thus cannot happen together
• If a list of events is mutually exclusive, it means that only one of them can possibly take place– Examples
• Toss one coin: E1 = {H}, E2 = {T}
• Toss two coins: E1: first toss is Head = {HT, HH}, E2: first toss is Tail = {TH, TT}
• Invest in a stock for one day: E1: One day stock returns greater than equal to 10% = {10% to +Infinity}, E2: One day stock returns between 2% to 5% = {2% to 5%}
– How would E1 compare with E3: One day stock returns less than or equal to 10% = {-100% to 10%)
• Lend money: E1: Borrower repays entire principal = {100,101,102,…112}, E2: Borrower repays 50% of principal = {50}
– How would E1 compare with E3: Borrower repays principal with 5% interest = {105} ?
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Properties of Probabilities defined
on a Sample Space• Sample space must be exhaustive: List all possible
outcomes
• Outcomes in the sample space must be mutually exclusive
• The exhaustive and mutually exclusive characteristic of a sample space together imply that
– The probability (likelihood of occurrence) of any one outcome or event must lie between 0 and 1
• 0 < P(E) <1
– The sum of the probabilities of all the outcomes in the sample space must be 1
• P(S) = 1
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Properties of Probabilities defined
on a Sample Space• By extension of the above rationale
– The probability of any event would lie between 0 (impossible) and 1 (certain), or 0 < P(E) < 1.
– There is no such thing as a negative probability (less than impossible?) or a probability greater than 1 (more certain than certain?).
– The sum of all probabilities of all outcomes would equal 1, provided the outcomes are both mutually exclusive and exhaustive.
– If outcomes are not mutually exclusive, the probabilities would add up to a number greater than 1, and if they were not exhaustive, the sum of probabilities would be less than 1.
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Classical / A Priori Approach to
Probabilities• When the range of possible uncertain outcomes in a
sample space is known and equally likely
• Probability for each outcome or event can be determined by logic
• Examples– Tossing a fair coin – outcomes can only be heads or tails and
both are equally likely
– Rolling a fair die – outcomes can only be 1,2…,6 and all six are equally likely
– Drawing a card from a pack – outcomes can only be 52, and all are equally likely
• The probability of each outcome in the above examples has been determined by construction of the coin/die/pack of cards
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Defining A Priori Probabilities
• In a sample of N mutually exclusive, exhaustive and equally likely outcomes we assign a chance (or weight) of 1/N to each outcome
• We define the probability of an event for such a sample as follows:
• The probability of an event E occurring is defined as:
• P(E) = n(E)/n(S)
– n(E) is the number of outcomes favourable to E and
– n(S) is the total number of equally likely outcomes in the sample space S of the experiment
• By extension, the probability of event E not occurring
• P(not E) = 1-P(E) = 1- n(E)/n(S)
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Example
• What is the probability that a card drawn at random from a deck of cards will be an ace?
• Total no. of outcomes in the sample space?
• Are they equally likely?
• Event of interest (E)?
• No. of Outcomes favourable to the Event?
• P(E)?
• What is the probability that a card drawn at random from the deck will not be an ace?
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A Priori Probabilities
• The same principle can be applied to the problem of determining the probability of obtaining different totals from a pair of dice.
• Possible Outcomes?
• Are they equally likely?
• Event 1 = A = sum of the two dice will equal 5
• Event 2 = B = the absolute difference will equal 1
• P (Event 1) =
• P (Event 2) =
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A Priori Probabilities
• In certain cases where outcomes are not equally likely,
one can still deduce rationally the a priori probability of
an event
• For example
– If we forecast that a company is 70% likely to win a
bid on a contract (irrespective of how this probability
is derived), and we know this firm has just one
business competitor, then we can also make an a
priori forecast that there is a 30% probability that the
bid will go to the competitor
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Empirical Probabilities
• In finance, we cannot depend upon the exactness of a process to determine a priori probabilities
• The financial analyst would have to depend upon historical occurrence of the event(s) or repeat an experiment multiple times to determine the probability of the event empirically
• For example, the range of outcomes of returns on a financial asset are virtually infinite and that too, not all outcomes are a priori, equally likely
• Thus, the financial analyst would have to observe many movements in asset prices to determine the probability of future price changes of a given magnitude
• Of course, we know that past performance does not guarantee future results, so a purely empirical approach has its drawbacks
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Defining Empirical Probabilities• The probability of an event (or outcome) is the proportion
of times the event occurs in a long run of repeated
experiment.
• The empirical probability of a given outcome / event Z is
defined as
– P(Z) = no. of Z occurrences/no. of trials of the experiment
• This is the same as analysis of relative frequency of
observations in a sufficiently large sample
• Consider a financial analyst who is interested in knowing
what will be the probable one day returns on a particular
stock.
– He tracks the past 100 days of price movements and returns of
a particular stock.
– Each of the 100 days would constitute trial and each day’s
returns would constitute the outcome
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Subjective Probabilities
• Probability under this approach is simply defined as the strength of belief that an event will occur
• It is based upon experience and judgment
• Such probabilities are applied to many business problems where a priori probabilities are not possible, nor are there sufficient empirical observations upon which to base probability estimates
• For example, subjective probability is incorporated in the forecasting of company profits by investment analysts
• Of course, subjective probabilities are unique to the person making them and to the specific assumptions made
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Rules of Probability
• There are a number of formal probability rules applied to
probability estimates
• Which of these rules is applicable will depend upon
whether
– We are concerned with a single event, in which case
the outcomes relate only to that event
– We are concerned with combinations of several
events, for example the changes in Sensex and
Exchange Rates together
– The combined events are independent or mutually
exclusive
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Rules of Probability
• The rules are
– Complement rule: When we are concerned with
whether an event A will not occur
– Multiplication rule: when we are concerned with event
A and B occurring together. This requires us to know
whether A and B are independent of each other
– Addition rule: when we are concerned with event A or
B happening. This requires us to know whether A and
B are mutually exclusive
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Intersection of Events and Joint
Probability
• Joint probability: The probability of both event A and
event B occurring: P(A and B) / P(A ∩ B) / P(AB)
• Intersection: Event defined as both A and B occur
• Example Using a priori probabilities:
– P(A and B) = No. of Outcomes favourable to the Joint Event /
Total no. of outcomes in the sample space
– Event A is you draw a spade from a deck of cards
Event B is you draw a king
– Intersection is event you draw a king of spades
– Joint probability = P(A and B) = 1/52
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Joint Probability Table
Job_statusDecision
accept reject
governmen 0.116 0.072
military 0.003 0.003
misc 0.000 0.003
private_s 0.266 0.352
retired 0.014 0.005
self_empl 0.035 0.051
student 0.005 0.008
unemploye 0.013 0.056
• Interpretation of 0.266?
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Marginal Probability
• Marginal probability: Probability of a single event
• When outcomes are exhaustive and mutually exclusive, can be calculated by adding all the joint probabilities containing the single event
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Marginal Probability
Job_statusDecision
accept reject Total
governmen 0.116 0.072 0.187
military 0.003 0.003 0.005
misc 0.000 0.003 0.003
private_s 0.266 0.352 0.617
retired 0.014 0.005 0.019
self_empl 0.035 0.051 0.086
student 0.005 0.008 0.013
unemploye 0.013 0.056 0.069
Total 0.451 0.549 1.000
• Interpretation of 0.187, 0.451?
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Conditional Probability
• A conditional probability is the probability of an event given that another event has occurred
• Conditional probability assumes that one event has taken place or will take place, and then asks for the probability of the other (A, given B)
• P(B | A): probability of B given A
• P(A | B): probability of A given B
• For example, – Probability of drawing a king given that a spade is drawn: P(king
| spade) = 1/13
– Probability of drawing a spade given that a king is drawn: P(spade | king) = 1/4
– What is the probability that the total of two dice will be greater than 8 given that the first die is a 6?
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Conditional Probability in Terms of
Joint Probability
Job_statusDecision
accept reject Total
governmen 0.116 0.072 0.187
military 0.003 0.003 0.005
misc 0.000 0.003 0.003
private_s 0.266 0.352 0.617
retired 0.014 0.005 0.019
self_empl 0.035 0.051 0.086
student 0.005 0.008 0.013
unemploye 0.013 0.056 0.069
Total 0.451 0.549 1.000
•Given that an application
is accepted, what is the
probability that the
customer is a Pvt. Sector
employee?
•What kind of probability
asked for?
•Imagine 1000 Customer
Applications. You would
expect 451 to be accepted
and 266 of those to be
Pvt. Sector employees
•P(Pvt. S Empl | Accept) =
116/451 = 0.257
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Conditional Probability in Terms of
Joint Probability• Joint Probability can be rescaled to find Conditional
Probability
P(A|B) = P(A and B) / P(B)
• Can think about this as rescaling the Accept Column so it sums to one
• Interpretation– P(Pvt. S Empl | Accept) = 266/451 = 0.59– P(Non Pvt. S Empl | Accept) = 185/451 = 0.41
– Means Pvt. S Employees are more likely to be accepted than Non Pvt. Sector Employees?
– P(Accept | Pvt. S Empl) = 266/617 = 0.43
– P(Accept | Non Pvt. S Empl) = 185/383 = 0.483
– Means more of the accepted applicants are Non Pvt. Sector Employees than Pvt. Sector Employees?
– If the bank’s objective was to specifically target a group for marketing of the product, which probabilities would it rely upon to give it an accurate picture?
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Joint Probability in Terms of
Conditional Probability• Another way to calculate Joint Probability of Events A
and B isP(A and B) = P(A) x P(B|A) = P(B) x P(A|B)
• where P(B|A) is the conditional probability of B given A and P(A|B) is the conditional probability of A given B
• Example: – If we believe that a stock is 70% likely to return 15% in the next
year, as long as GDP growth is at least 8%, then we have made our prediction conditional on a second event (GDP growth). In other words, event A is the stock will rise 15% in the next year; event B is GDP growth is at least 8%; and our conditional probability is P(A | B) = 0.7
– Now if we know that there is a 20% unconditional probability that GDP will grow at 8% or above P(B) = 0.2
– The probability that GDP will grow at least at 8% and stock will return 15% is P(A and B) = 0.7 *0.2 = 0.14
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Joint Probability Vs Conditional
Probability• Joint probability is not the same as conditional
probability, though the two concepts are often confused.
• Joint probability sets no conditions on the occurrence of events but simply provides the chance that both events will happen together
• In a problem, to help distinguish between the two, look for qualifiers that one event is conditional on the other (conditional) or whether they will happen concurrently (joint).
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Joint Probability and Independence
of Events• Independence of Event A and B: When Probability of
Event A occurring does not depend upon occurrence of Event B and vice versa
• Two events are independent if and only if
• P(A | B) = P(A) [implies P(B | A) = P(B)]
• If A and B are independent, then the probability that events A and B both occur is:P(A and B) = P(A) x P(B | A) = P(A) x P(B)
• Examples:– What is the probability that a fair coin will come up with heads
twice in a row?
– Now consider a similar problem: Someone draws a card at random out of a deck, replaces it, and then draws another card at random. What is the probability that the first card is the ace of clubs and the second card is a club (any club)?
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Joint Probability and Independence
of Events• Is the customer application being accepted independent
of being a Pvt. sector employee?
• P(Accept | Pvt. Sector employee) = 0.43
• P(Accept) = 0.451
• The rule generalizes for more than two events provided they are all independent of one another, so the joint probability of three events P(ABC) = P(A) * (P(B) * P(C), again assuming independence
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Joint Probability and Mutually
Exclusive Events• The joint probability of two mutually exclusive events
occurring is 0
• This is because by definition of mutually exclusive, events A and B cannot occur together
• For example:– Roll a die once: Event A = 6, Event B <5
– Are A and B mutually exclusive?
– P(A and B) = ?
– Roll 2 dice: Event A = {1,4}, Event B = {4,1}
– Are A and B mutually exclusive?
– P(A and B) = ?
– Roll 2 dice: Event A = {1,4}, Event B = at least one die shows 1
– Are A and B mutually exclusive?
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Union of Events and Probability
• Union: Union of events A and B is event that either A or B or both occur: (A or B) or (AUB)
• P(A or B) is interpreted as the probability that at one of the two events A and B will occur
• If events A and B are mutually exclusive, P(A and B) = 0
• P(A or B) = P(A) + P(B)
• Example:– What is the probability of rolling a die and getting either a 1 or a
6?
– Since it is impossible to get both a 1 and a 6, these two events are mutually exclusive
– P(1 or 6) = P(1) + P(6) = 1/6 + 1/6 = 1/3
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Union of Events and Probability
• If events A and B are not mutually exclusive, P(A and B) ≠ 0
• P(A or B) = P(A) + P(B) – P(A and B)
• The logic behind this formula is that when P(A) and P(B) are added, the occasions on which A and B both occur are counted twice. To adjust for this, P(A and B) is subtracted.
• Example:– What is the probability that a card selected from a deck will be
either an ace or a spade?
– P(Ace) = 4/52
– P(Spade) = 13/52
– P(Ace and Space) = ?
– P(Ace or Space) = ?
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Union of Events and Probability
• Consider the probability of rolling a die twice and getting a 6 on at least one of the rolls. The events are defined in the following way:
• Event A: 6 on the first roll: P(A) = 1/6
• Event B: 6 on the second roll: P(B) = 1/6
• P(A and B) = 1/6 x 1/6 (why?)
• P(A or B) = 1/6 + 1/6 - 1/6 x 1/6 = 11/36
• The same answer can be computed using the following admittedly convoluted approach:
• Getting a 6 on either roll is the same thing as not getting a number from 1 to 5 on both rolls.
• This is equal to: 1 - P(1 to 5 on both rolls) = 1 – 5/6 x 5/6 = 1 – 25/36 = 11/36
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Union of Events and Probability
• Despite the convoluted nature of this method, it has the advantage of being easy to generalize to three or more events.
• For example, the probability of rolling a die three times and getting a six on at least one of the three rolls is?
1 - 5/6 x 5/6 x 5/6 = 0.421
• In general, the probability that at least one of k independent events will occur is:
1 - (1 - α)k
where each of the events has probability α of occurring
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Union of Events and Probability
• Example:
• Assume that the bank had lent to 5 different borrowers, each of whom was assigned a probability of default = 0.1
• The bank wished to hedge the risk of its portfolio such that it bought insurance such that the loss against at least one borrower defaulting was made good
• How would the insurance company estimate the probability of at least one borrower defaulting, assuming that the default event of each borrower was independent?
• P(At least one default) = 1 – (1-0.1)5 = 0.41
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Union of Events and Probability
Job_statu
s
Decision
accept reject Total
governme
n 0.116 0.072 0.187
military 0.003 0.003 0.005
misc 0.000 0.003 0.003
private_s 0.266 0.352 0.617
retired 0.014 0.005 0.019
self_empl 0.035 0.051 0.086
student 0.005 0.008 0.013
unemploy
e 0.013 0.056 0.069
Total 0.451 0.549 1.000
•What is the probability that the
customer application is accepted
or the customer is a government
employee?
•What kind of probability asked
for?
•P(Accept) = 0.451
•P(Govt. Employee) = 0.187
•P(Accept and Govt. Employee =
0.116
•P(Accept or Govt. Employee) =
0.451 + 0.187 – 0.116 = 0.522
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Union of Events and Probabilities
• Example:
• A Fund manager has invested in the stocks of two companies 1 and 2 and is interested in knowing what is the probability that the equity price of either company will rise
• Are the two events mutually exclusive?
• P(Co. 1 Equity ↑) = 0.55
• P(Co. 2 Equity ↑) = 0.35
• P(Co. 1 Equity ↑ and Co. 2 Equity ↑) = 0.3
• P(Co. 1 Equity ↑ or Co. 2 Equity ↑) = 0.55+0.35-0.30 = 0.60
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Union of Events and Probability
• What if we want to know the probability of at least one
of 3 events A, B and C happening?
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )P A B C P A P B P C P AB P AC P BC P ABC
A B
C
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Probability Rules Summary
• Multiplication Rule: Joint probability of any two events A and B is:– P(A and B) = P(A | B) * P(B)
– Follows from definition of conditional probability
• Multiplication Rule Independent Events: If A and B are independent, joint probability is:– P(A and B) = P(A) * P(B)
• Addition Rule mutually exclusive events:– P(A or B) = P(A) + P(B)
– Mutually exclusive if both cannot occur
• Addition Rule: Probability that event A or event B or both occur is– P(A or B) = P(A) + P(B) – P(A and B)
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• Are mutually exclusive events independent?
• Does pr(A|B) = pr(A)?
• NO!
• If B has happened, then A can not happen
• pr(A|B) = 0
• So
• Mutually exclusive events are DEPENDENT.
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Probability Tree and Total
Probability Rule
Job_statusDecision
accept reject
Private
Sector 0.29 0.35
Non Pvt
Sector 0.16 0.20
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Probability Tree
Accept: 0.45
Reject: 0.55
Pvt. Sector | Accept: 0.59
Not Pvt. Sector | Accept: 0.41
Pvt. Sector | Reject: 0.64
Not Pvt. Sector | Reject: 0.36
0.29
0.16
0.35
0.20
Pvt. S & Accept
Not Pvt. S & Accept
Pvt. S & Reject
Not Pvt. S & Reject
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Total Probability Rule
• The Total Probability Rule: The total probability rule explains an unconditional probability of an event A, in terms of that event's conditional probabilities in a series of mutually exclusive, exhaustive scenarios of event B
• P(A) = P(A | B) x P(B) + P(A | not B) x P(not B)
• With the total probability rule, event A has a conditional probability based on each scenario of event B (i.e. the likelihood of event A, given that scenario), with each conditional probability weighted by the probability of that scenario for event B occurring
• Example: – What is the probability of a customer employed in the Private
Sector:
– P(Pvt. S. Empl) = P(Pvt. S Empl | Accept) x P(Accept) + P(Pvt. S Empl | Reject) x P(Reject) = 0.59*0.45+0.64*0.55 = 0.61
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Total Probability Rule
• A model that predicts whether a borrower has defaulted or not is 95 percent effective in predicting that a borrower has defaulted when it actually has. However, the model also yields a “false positive” result for 1 percent of the non-defaulted borrowers. That is, there is 1 percent chance that a borrower who has not defaulted will be identified as a defaulted borrower by the model.
• Q: If 0.5 percent of the bank’s portfolio has actually defaulted, what is the probability that a borrower has defaulted given that the model predicts default?
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Probability Tree
Default: 0.005
No Default: 0.995
Def Pred | Default: 0.95
No Def Pred | Default: 0.05
Def Pred | No Default: 0.01
No Def Pred | No Default: 0.99
0.00475
0.00025
0.00995
0.98505
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Total Probability Rule And Baye’s
Theorem
• Let D be the event that a borrower has defaulted
• Let E be the event that the model predicts default
• We need to estimate P(D|E)
• We know: P(E|D)=0.95, P(E|notD)=0.01, P(D)=0.005,
P(notD)=1-P(D)=0.995
0147.0995.001.0005.095.0)()()()()(
00475.0005.095.0)()()(
3231.00147.0
00475.0
)(
)()(
notDPnotDEPDPDEPEP
DPDEPEDP
EP
EDPEDP