This document provides an overview of key concepts in probability and examples to illustrate those concepts. It discusses probability distributions, expected value, standard deviation, discrete uniform and binomial distributions. It also includes examples demonstrating conditional probability, Bayes' theorem, independence of events, and the Monty Hall problem. Practice problems are provided to help students apply these probability concepts.

1. 7/2/2012
Introduction to Probability:
Quantitative Methods – I
Bangalore
[QM 105]
Sessions 3-4-5: price range (lakh frequency
Review Probability/ practice problems <=10 17
Bayes Theorem 10 to 20 46
Upgrading prior probabilities on Market Share 20 to 30 40
Bayesian vs. Frequentist 30 to 50 47
Puzzles 50 to 75 25
How reliable is RELIABLE? 75 to 100 13
Clear Tone Radios 100 to 135 9
Characteristics of discrete probability distribution > 135 3
Expected value, standard deviation of a random variable
Discrete Uniform and Binomial Probability Distributions
price range (lakh frequency
<=10 17
Area (sq ft) frequency
Relative Frequency to 10 to 20
20 to 30
30 to 50
46
40
47
<= 600
600 to 800
15
20
Probability 50 to 75
75 to 100
100 to 135
25
13
9
800 to 1000 35
> 135 3
1000 to 1250 70
1250 to 1500 20
One flat is chosen at random in Bangalore. What is the probability
1500 to 2000 15
that its price is
> 2000 25
What is the probability that the chosen flat has area > 2000 sqft?
• 10 lakh or less? 0.085
0.315 0.125
• 20 lakh or less?
• More than 20 lakh? 0.685
What is the probability that chosen flat has costs no higher than
• Between 10 and 100 lakh? 0.855 a crore but has area more than 2000 sqft?
0.125 ×0.94 ?
Joint Probability
Area (sq ft)
<= 600 to 800 to 1000 to 1250 to 1500 to > Margi
Area (sq ft)
600 800 1000 1250 1500 2000 2000 nal
<= 600 to 800 to 1000 to 1250 to 1500 to > Margin
600 800 1000 1250 1500 2000 2000 al <=10 0.07 0.015 0 0 0 0 0 0.085
<=10 14 3 17 10 to 20 0.005 0.085 0.145 0 0 0 0 0.23
10 to 20 1 17 29 46
20 to 30 0 0 0.03 0.17 0 0 0 0.2
20 to 30 6 34 40
price
price 30 to 50 0 0 0 0.18 0.055 0 0 0.235
30 to 50 36 11 47 range
rang (lakh)
50 to 75 9 13 3 25 50 to 75 0 0 0 0 0.045 0.065 0.015 0.125
e
(lakh 75 to 100 1 12 13 75 to 100 0 0 0 0 0 0.005 0.06 0.065
100 to
100 to 135 0 0 0 0 0 0.005 0.04 0.045
135 1 8 9
> 135 3 3 > 135 0 0 0 0 0 0 0.015 0.015
Marginal 15 20 35 70 20 15 25 Marginal 0.075 0.1 0.175 0.35 0.1 0.075 0.125 1
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Conditional Probability Conditional Probability
Given that a flat has more than 2000 sq ft, what is the probability that
it costs more than 100 lakh?
P[ A and B] P ( A ∩ B)
P[ A given B] = P ( A | B) =
P[ B ] P( B )
Area > 2000
Price≤100
area ≤ 2000
price> 100
Classical approach:
(In) dependence of Events Counting argument
Q. A team of 5 members is to be selected randomly
• A and B are said to be independent if from 6 gentlemen and 4 ladies. What is the probability
– P[A|B] = P[A] that the team will have
– or equivalently P[A|B] = P[A|not B] or...
– or equivalently P[A and B] = P[A] × P[B]
a) No ladies?
b) Three gentlemen?
Connection between independent and disjoint events c) At most three gentlemen?
Assumption: all possible selections are equally likely
Problem (Easy) Problem (easy)
The HAL Corporation wishes to improve the resistance of its personal computer to
disk-drive and keyboard failures. At present, the design of the computer is such that
At a soup kitchen, a social worker gathers the following data. disk-drive failures occur only one-third as often as keyboard failures. The probability
Of those visiting the kitchen, 59 percent are men, 32 percent are of simultaneous disk-drive and keyboard failure is 0.05.
alcoholics, and 21 percent are male alcoholics. What is the probability
that a random male visitor to the kitchen is an alcoholic? (a) If the computer is 80 percent resistant to disk-drive and/or keyboard failure,
how low must the disk-drive failure probability be?
(b) If the keyboard is improved so that it fails only twice as often as the disk-drive
P[alcoholic and male] (and the simultaneous failure probability is still 0.05), will the disk-drive failure
P[alcoholic given male]= probability from part (a) yield a resistance to disk-drive and/or keyboard failure
P[male] higher or lower than 90 per cent?
0.21
= = 35.59%
0.59
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Solution Use of probability in questionnaire design:
How to get an answer w.o. being sure you’ve asked the question
• Flip a coin. If you get H answer Q1 ONLY(in yes/no form) If you
D → disk-drive failure
K → keyboard failure get T answer to Q2 only (in yes/no form)
P(K ∩ D) = 0.05 P(K) = 3 P(D) = 3x say • Q1. Is your mother born in May?
• Q2. Do you find this class useless?
a) P(K ∪ D) =0.2 = x + 3x –0.05 ⇒ x = 0.0625 = P(D)
b) If P(K) = 2 P(D) = 0.125, then P(K ∪D) = 0.1375 P[' yes' ] − P[Q1] × P[' yes' | Q1]
and P[ (K ∪D)c ] = 0.8625 < 90% P[' yes' | Q 2] =
P[Q 2]
i.e, the computer is only 86.25% resistant to failure of either type
Finding your perfect “match”
Which category of students do you
No. Criterion % of opposite
gender who satisfy belong to?
the criterion
1 P(ready in
P( not
P(not ready P(category
students % of ready in
2 category students
each
each
and | not
category) catgeory..) ready)
3 category)
A 40.00% 0.9 0.1 0.04 12.12%
N
B 35.00% 0.6 0.4 0.14 42.42%
The Chain-rule C 25.00% 0.4 0.6 0.15 45.45%
P[ B1 and B2 and B3 ] =P[B1 ] × P[ B2 | B1 ] × P[ B3 | B1and B2 ]
p(not
(if independent) =P[B1 ] × P[ B2 ] × P[ B3 ] ready) 0.33
Bayes Rule
Bayes Rule -- using table
P[ A]× P[B | A] events Prior Prob P[B|Ai] P[BAi] P[Ai|B]
P[ A | B] = P[PA∩]B] =
[B
P[Ai]
(ii)
(iii)
=(i)*(ii)
(iv)
=(iii)/P[B]
P[ A ∩ B] + P[ B ∩ Ac ] A1
(i)
P[A1] P[B|A1] P[A1B] P[A1| B]
A2 P[A2] P[B|A2] P[A2B] P[A2| B]
P[ A]× P[ B | A] Ak P[Ak] P[B|Ak] P[AkB] P[Ak| B]
=
P[ A] × P[B | A] + P[ Ac ]× P[ B | Ac ] Sum 1 P[B]
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Monty Hall Problem
or Story of a father
‘Khul Ja SimSim’
A father announces: “I have two children, born
• The car is behind one of the 3 doors.
three years apart, one of whom is a boy.”
• You select door A. What is the probability that the other child is a
• Aman Verma (who knows where the car is) boy?
opens door B and shows that this is empty.
Gives you an option of “switch”.
• Should you stick to your initial choice?
“Information – the new language of science”
by Hans Christian von Baeyer.
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