1. MNGT 501 (part 2) Roger Brooks

2. MNGT 501
Last term we covered sampling, describing categorical data, describing numerical data, basic maths
This term the lectures will cover:
Week 12: Probability
Week 13: Expected values, probability distributions
Week 14: Correlation and regression
Week 15: Revision

3. MNGT 501 There will be workshops in weeks 12 – 14
Related to the lecture that week
2 hour multiple choice exam
Friday 14th February 2.30 p.m. (see details on Moodle)
The mark for this will be your course mark
It will include both lecture and workshop material
Closed book with formula sheet

4. Probability

5. Probability and business
You have sales visits to 2 major customers. What is the chance that at least one will place an order?
You run an events company and are planning an an outdoor weekend event. What is the probability that it will rain on both days?
What is the chance that none of the machines on the production line will break down today?

6. Introduction Probability means: chance that event happens
Games / gambling
Sports events
Business
product being successful
effect of an advertising campaign
obtaining a particular contract
number of customers at a shop in the next hour
project being finished on time
adverse / beneficial legislation
production line failures
staff shortages
travel delays
value of an investment increasing

7. Interpreting probability values
If p is the probability of an event:
0  p  1
p = 0 means the event cannot occur
p = 1 means the event is certain to occur
The closer to 1, the more likely the event
p = 0.1
p = 10%
p = 1/10
1 in 10 chance
Never happens 1
Always happens
Dice: Throw Throw 1 or 2 or 3 or 4 or 5 or 6

8. Probability: obtaining values
A priori – based on the assumptions of the situation
Relative frequency – based on historical data
Subjective – based on expert opinion
Roll a dice: P(roll a 2) =
(assume each number equally likely)
P(Messi score penalty) = no. successes
no. attempts
P(Labour win next election) =

9. Law of Large Numbers
For a repeated event: in the very long run the probability is the proportion of times we expect the event to occur
But: this does not enable us to predict the next outcome!
Dice: expect to get a 2 in about 1/6 of rolls

10. Definitions Outcome (or elementary event)
Something that might happen and that cannot be sub-divided into separate occurrences
Sample space
Set of all possible outcomes for the situation of interest
Event
A set of one or more outcomes
Throw 3
S = {1, 2, 3, 4, 5, 6}
Even number = {2, 4, 6}

11. Venn diagram representation
P(S) = 1
0 ≤ P(A) ≤ 1
In the diagram: area shows probability and so P(A) is the proportion of S taken up by A

12. Complementary events P(A) + P(not A) = 1 Hence, P(A) = 1 - P(not A) S
E.g., dice:
P(throw 3) = 1/6
P(not 3) = 1 – 1/6 = 5/6
P(A) + P(not A) = 1
Hence, P(A) = 1 - P(not A)
It is sometimes easier the work out the probability that the
event we are interested in doesn’t happen.
See later: instead of “at least one” calculate the probability of none.

13. Rules for combining events
If we know the probabilities of two events A and B, can we calculate combined probabilities?
A or B
A and B

14. Venn diagram for combining events
A or B A  B Union
A and B A  B Intersection

15. P(A or B): addition rule

16. Disjoint / mutually exclusive
A and B are mutually exclusive or disjoint means: A and B can’t both happen [ i.e., P(A and B) = 0] S A B

17. Addition rule for disjoint / mutually exclusive eventsB
P(A or B) = P(A) + P(B)

18. Mutually exclusive example 1
What is the probability of throwing either an even number or a 5?
A = even number = {2, 4, 6}: P(A) = 3/6
B = throw 5 = {5}: P(B) = 1/6;
A and B = { }: P(A and B) = 0/6
P(A or B) = P(A) + P(B)
= 3/6 + 1/6 = 4/6
Check:
A or B = {2, 4, 5, 6}: P(A or B) = 4/6

19. Mutually exclusive example 2
The probability that a plastic component is too large is 1/8. The probability that it is too small is 1/6. What is the probability that a randomly chosen component is either too large or too small?

20. General addition rule Note: “A or B” means: A only or B only or both   
P(A or B) = P(A) + P(B) – P(A and B) S A B

21. Simple addition rule example (dice)
What is the probability of throwing either an even number or a number less than 4?
A = even number = {2, 4, 6}: P(A) = 3/6;
B = less than 4 = {1, 2, 3}: P(B) = 3/6;
A and B = {2}: P(A and B) = 1/6
P(A or B) = P(A) + P(B) – P(A and B)
= 3/6 + 3/6 – 1/6 = 5/6
Check:
A or B = {1, 2, 3, 4, 6}: P(A or B) = P{1, 2, 3, 4, 6} = 5/6

22. Addition rule example 1
In an ordinary pack of cards (no jokers) what is the probability of drawing an ace or a spade?

23. Addition rule example 2
Two tasks in a construction project are building the structure and laying the service pipes.
The probability of the structure being late is 3/10
The probability of the pipes being late is 2/5
The probability of both being late is 1/5
What is the probability that the structure or the pipes will be late?

24. Mutually exclusive is a special case of the general rule
P(A or B) = P(A) + P(B) – P(A and B) Mutually exclusive by definition means: P(A and B) = 0 Hence, for mutually exclusive events: P(A or B) = P(A) + P(B)

25. P(A and B): multiplication rule

26. Independent events A and B are independent means:
whether or not A occurs has no effect on the chance of B occurring (and vice versa).
E.g., toss a coin twice.
A = head on 1st, B = head on 2nd.
Independent because probability of B doesn’t depend on whether A happened:
P(B) = 1/2 whether we get head or tail with 1st toss

27. Multiplication rule for independent eventsB
P(A and B) = P(A) × P(B)

28. Independence example
Toss a coin twice. What is the probability of 2 heads?
A = head on 1st; B = head on 2nd. Independent
P(A) = 1/2, P(B) = 1/2
P(A and B) = P(A) × P(B) = 1/2 × 1/2 = 1/4
Check: st nd
T T
T H
H T
H H

29. Effect on addition rule for independent events
P(A or B) = P(A) + P(B) – P(A and B)
For independent events: P(A and B) = P(A) × P(B)
So, P(A or B) = P(A) + P(B) – P(A) × P(B)

30. Independence example 1
A company has an important order to manufacture next week. The order requires two components, A and B, to be delivered in advance by two separate companies. The chance of component A arriving this week is 5/6. The chance of component B arriving this week is 1/2.
What is the chance that:
a) A and B arrive in time
b) A or B arrive in time

31. Independence example 1
a) What is the chance that A and B arrive this week?
b) What is the chance that A or B arrive this week? 31

32. Independence example 2
A production line has 3 machines in series. The probability each day that they breakdown is 1/10, 1/25 and 1/50 respectively. What is the probability that at least one of the machines breaks down on a given day?
“At least one” so calculate probability that all working 32

33. Independence example 2
Probability they breakdown is 1/10, 1/25 and 1/50 respectively. What is the probability that at least one of the machines breaks down?
“At least one” so calculate probability that all working
M M M3
P(breakdown) / / /50
P(works o.k.) / / /50
Assume independence:
P(all work) = P(M1 works) × P(M2 works) × P(M3 works)
= 9/10 × 24/25 × 49/50 = 10584/12500
P(at least one breaks down)
= 1 – P(all work) = 1 – 10584/12500 = 1916/12500 = 0.15 33

34. Conditional probability
Definition: “B | A” means “B given A”
P(B | A) is the probability that B will happen if A has already happened
P(B | A) = P(A and B) / P(A) S
A has happened and so our sample space effectively reduces to A A B

35. General multiplication rule
Rearrange the equation for conditional probability:
P(B | A) = P(A and B) / P(A)
Hence for any events A and B:
P(A and B) = P(A) × P(B | A) S
Adding more
conditions,
so less likely A B

36. Independent events is special case of general rule
If events are independent then P(B) is not affected by whether A has occurred. So,
P(B | A) = P(B)
So, for independent events:
P(A and B) = P(A) × P(B | A) = P(A) x P(B)

37. Multiplication rule example 1
In a group of 100 people there are 40 male and 60 female. 25 of the group live in Manchester (8 male and 17 female). If we choose one person at random what is the probability they are a male living in Manchester?
A = Male; B = Mcr
P(A) = 0.40; P(B) = 0.25
Obviously, the correct
answer is 0.08.
However, the example
illustrates how the
equation works.
But, P(B | A) = 8/40 = 0.20
P(A and B) = P(A) x P(B | A)
= 0.40 x 0.20 = 0.08

38. Multiplication rule example 2
A bag contains one red ball and two blue balls. If 2 balls are drawn without replacement what is the probability of choosing 2 blue balls?

39. Multiplication rule example 3
We are planning a big outdoor event over a weekend next April. Analysis of weather records for previous years show that:
it rains on 1/3 of the days in April
if it rains on one day then the chance of it raining on the next day is 1/2
What is the chance that it rains on both days?

40. Summary of equations Complementary events P(A) = 1 – P(not A)
A or B (addition rule)
P(A or B) = P(A) + P(B) – P(A and B)
If mutually exclusive: P(A or B) = P(A) + P(B)
A and B (multiplication rule)
P(A and B) = P(A) × P(B | A)
If independent: P(A and B) = P(A) × P(B)

41. Addition and multiplication
A or B:
More options therefore probability must get larger – hence add
A and B
More restrictive therefore probability must get smaller – hence multiply

42. Project bidding example
We have put in bids for two separate projects. The probabilities that the bids will be successful are estimated to be 5/8 and 3/4 respectively. What is the probability that both bids are successful? What is the probability that at least one of the bids is successful?

43. Construction example
Two tasks in a construction project are building the structure and laying the service pipes.
The probability of the structure being late is 3/10
The probability of the pipes being late is 2/5
The probability of both being late is 1/5
Are these events independent?

44. Airline example
Three major airlines are deciding whether to order planes from European manufacturer A or U.S. manufacturer B.
The probability of airline 1 ordering from A is 0.25 The probability of airline 2 ordering from A is 0.70 The probability of airline 3 ordering from A is 0.50 Assume that the ordering decisions of the airlines are independent events.
What is the probability that at least one airline orders from A?
What is the probability that exactly one airline orders from A?

45. Airline example
The probability of airline 1 ordering from A is 0.25 The probability of airline 2 ordering from A is 0.70 The probability of airline 3 ordering from A is 0.50
What is the probability that at least one airline orders from A?

46. Airline example
Probability of ordering is 0.25, 0.70 and 0.50 What is the probability that exactly one orders?

47. Health screening example
A health screening test is carried out on men aged 55 to test for a particular disease. Past data on the test indicates that:
if a man aged 55 has the disease then the probability of the test being positive is 0.90
if a man aged 55 does not have the disease then the probability of the test being negative is 0.85
The probability of a man aged 55 having the disease is If a 55 year old man tests positive what is the probability that he has the disease?

48. Health screening example
10 × 0.9 = 9
10 – 9 = 1
Assume 1000 people
990 × 0.85 = 841.5
990 – = 148.5
Disease
No disease
Total
Positive Negative Total 9 1 10
148.5
841.5
990
157.5
842.5
1000
Probability (Disease | Test positive ):
= 9 / = = 6% (to nearest percent) 48

49. Flight baggage example
Melanie is flying from city A to city C with a connection in city B. The probability that her first flight arrives on time is 0.3. If the flight is on time, the probability that her luggage will make the connecting flight is 0.9, but if the flight is delayed the probability that the luggage will make it is only 0.6. In either case, Melanie makes the flight.
If Melanie’s luggage is not there to meet her in city C (i.e., it didn’t make the connecting flight), what is the probability that the first flight was late in arriving in city B (to 2 decimal places)?

50. Probability - light bulbs example
In a box of 10 light bulbs, 4 are defective. If we choose 3 in turn and test them, what is the probability that:
The first 2 are defective and the third is o.k.?
At least one is defective?
General approach:
We require a certain sequence of events on the 3 picks (A and B and C), hence multiplication rule
Events are dependent – what happens on each pick affects the probabilities on the next pick:
P(A & B & C) = P(A) x P(B | A) x P(C | A & B)

51. Light bulbs example The first 2 are defective and the third is o.k. Ok1 2 3
Defective
We just need to keep track of which are left after each pick
1st pick: Defective
Probability = 4/10 = 2/5
There are now 9 left: 6 o.k. & 3 defective
2nd pick: Defective
Probability = 3/9 = 1/3
There are now 8 left: 6 o.k. & 2 defective
3rd pick: O.k.
Probability = 6/8 = 3/4
Hence, probability = 2 x 1 x 3
= 2 x 1 x 3
5 x 3 x 4
= 6 = 1 .

52. Light bulbs example
At least one is defective

53. Space Shuttle Risk assessments on solid booster rockets:
Historical: 1 failure per 57 firings
NASA internal study (1985): 1 in failure rate
External studies:
1981 study: 1 in 1000 failure rate
1983 study: 1 in 100 failure rate
1984 study: 1 in 210 failure rate
The shuttle has 2 rockets and failure of either will result in a crash
Shuttle was planned for 500 launches

54. Space Shuttle
The estimated failure rate of a shuttle booster rocket is 1/210. The shuttle has 2 rockets and both must work properly for a successful launch.
a) What is the probability of a successful launch?
b) In 500 launches, what is the probability of at least one crash?

55. Space Shuttle a) b) P(rocket o.k.) = 209/210
P(success) = P(1st o.k. AND 2nd o.k.)
= P(1st o.k.) x P(2nd o.k.)
= 209/210 x 209/210 =
P(at least 1 crash) = 1 - P(no crashes)
= 1 - P(1st o.k. and 2nd o.k. and ….)
= x x … = =

56. Space Shuttle
Unfortunately there was a crash: Challenger disaster in 1986 caused by booster rocket failure
Presidential Commission investigating the accident included famous physicist Richard Feynman
Richard Feynman severely criticised NASA’s internal estimate saying that “NASA exaggerates the reliability of its product, to the point of fantasy”
Any calculation / model is only as good as the data put into it