Probability Basics

DECS 430-ABusiness Analytics I

Reality

v There is rarely a professor or textbook handing us the data or the probabilities that we need to make a decision.

v Probabilities can be subjective OR objective but, outside of a casino, are rarely “knowable.”

v BUT we have to make a decision at some point. Where do we begin?

Using data: a simple example

You are in charge of the company’s charity golf event.

v On the (par 3) 7th hole, anyone who scores a hole-in-one wins a Porsche Cayenne S.

v Each of the 100 participants gets only one chance.v A company is offering you an insurance policy against

having to pay a winner. They charge $1167 for the policy.

Should you buy the insurance policy?

Hole-in-One insurance from US Hole In OneHoleinoneinsurance.com

Data from Golf Digest

What if we don’t have any data?

You are in charge of the Chicago Bulls’ halftime promotion –

v At every home game this year, one randomly selected fan will have the chance to win $1,000,000.

v All he/she has to do it make a ¾ court length shot during the halftime break.

v The fan gets only one chance…

v A company is offering you an insurance policy for $XXX. Should you buy the insurance policy?

Forecasting: What are the odds?

We sometimes use past data probabilistically: to assess the likelihood of future events.

What is the probability…

v … that my next customer is a parent? (Marketing)

v … that we will ever get paid for invoice #53498? (Accounting)

v …that Motorola will default on bond payments in 2016? (Finance)

v …that doubling our inventory will completely eliminate product shortages over the next year? (Operations)

v …that an amateur golfer hits a hole-in-one on a 165-yard hole?

How can airline companies keep customers loyal?

u Frequent flier programs (easy to implement, easily imitated)u Food (harder to achieve, but also harder for competitors to imitate)

From the perspective of a catering company (LSG Sky Chefs), imagine that you are in charge of redesigning the kinds of menu offerings that will be made on international flights. One of the general issues you face is whether to move toward healthier food or toward tastier food.

u What data could/would you gather?u How would you structure the information in the press release?

LSG Sky Chefs

LSG Sky Chefs

LSG Sky Chefs provides meals to airlines. Supplying roughly a million meals per day to the airline industry, this company certainly cares about the food tastes of passengers.

Some years ago, LSG Sky Chefs issued a press release reporting on a survey they conducted on this topic. We’ll look at an edited version of that actual press release. The category names have not been altered!

Frequent Flyers Reveal Recipe for Success When It Comes to Airline Food

NEW YORK (BUSINESS WIRE) The results of a national study of frequent flyers from LSG Sky Chefs, the world's largest airline caterer, show that U.S. airlines may be missing an opportunity when it comes to feeding their passengers.

The study identified, for the first time, four consumer segments distinguished by their attitudes and preferences for food service.

Four consumer segments distinguished by their attitudes and preferences for food service:

14% are the HUNGRY, HEALTHY FOOD FOLLOWERS. They like to eat full meals on airplanes and favor healthy, vegetarian dishes. Of the HHFFs: 75% of this segment said that food service is important to them on international flights.

20% are the HEALTH FOOD NIBBLERS. They are also characterized as health-food oriented but do not prefer large meals on an airplane. Of the HFNs: 53% of this segment said that food service is important to them on international flights.

31% are the BLUE SUIT GOURMANDS. This group consists of frequent flyers who have the highest average income. This is the group that takes airline food the most seriously and is willing to pay more for it. Of the BSGs: 89% of this segment said that food service is important to them on international flights.

35% are the NAYSAYERS, who are the polar opposites of the Blue Suit Gourmands in that healthy foods are not of particular importance and do not see a difference in food service among airlines. Of the Ns: 30% said that food service is important to them on international flights.

LSG Sky Chefs

Four consumer segments distinguished by their attitudes and preferences for food service:

14% are the HUNGRY, HEALTHY FOOD FOLLOWERS. They like to eat full meals on airplanes and favor healthy, vegetarian dishes. Of the HHFFs: 75% of this segment said that food service is important to them on international flights.

20% are the HEALTH FOOD NIBBLERS. They are also characterized as health-food oriented but do not prefer large meals on an airplane. Of the HFNs: 53% of this segment said that food service is important to them on international flights.

31% are the BLUE SUIT GOURMANDS. This group consists of frequent flyers who have the highest average income. This is the group that takes airline food the most seriously and is willing to pay more for it. Of the BSGs: 89% of this segment said that food service is important to them on international flights.

35% are the NAYSAYERS, who are the polar opposites of the Blue Suit Gourmands in that healthy foods are not of particular importance and do not see a difference in food service among airlines. Of the Ns: 30% said that food service is important to them on international flights.

LSG Sky Chefs

What is the probability that a randomly chosen survey participant…v … is in either the HHFF or HFN category?

u P(HHFF or HFN) = 14% + 20% = 34%

v … is in BSG category and thinks food is important?u P(BSG and YES) = 27.6%

v … thinks food is important?u P(YES) = 10.5% + 10.6% + 27.6% + 10.5% = 59.2%

v …from the BSG category thinks food is important?u P(YES | BSG) = 89%

LSG Sky Chefs

Spotting conditional probability

v The probability Motorola never defaults on its 10-year bondu given that Motorola hasn’t defaulted as of 2015.

v The probability that my next customer has childrenu given that he is a male in the age category 21-25.

v The probability that we will ever get paid for invoice #53498u given that payment is already 3 months past due.

Learning that one event has happened may change the perceived probability of another event occurring.

ü

ü

If someone likes healthy food, how likely is it that they think food is important on international flights?

We care about the events:

A = “likes healthy food” (HHFF or HFN)

B = “Yes” (thinks food important on int’l flights)

P(B | A) = P(A and B) / P(A) = (0.105 + 0.106)/(0.340) = 62.1%

LSG Sky Chefs

ü

ü

LSG Sky Chefs

If someone thinks food is important on international flights, how likely is it that they want healthy food?

We care about the events:

A = “likes healthy food” (HHFF or HFN)

B = “Yes” (thinks food important on int’l flights)

P(A | B) = P(A and B) / P(B) = (0.105 + 0.106)/(0.592) = 35.6%

LSG Sky Chefs

An entire reversed (“flipped”) tree can be built this way.

This is probably the way the data should have been presented in the original release:

“Of those passengers for whom eating is an important component of the flight experience, the largest group (nearly half of all fliers) want tasty, enjoyable meals. The remainder splits pretty evenly between those wanting healthy meals, those wanting healthy snacks, and those who just don’t care.”

Bayes’ Rule

v Any complete probability tree can be “flipped.”

v Alternatively, one can apply the definition of conditional probability, or one can apply Bayes’ Rule:

P(B | A) = P(AB) / P(A)

= P(AB) / [P(B)*P(A | B) + P(not B)*P(A | not B)]

v You may use whichever is more convenient. (I personally prefer the tree approach.)

Motivating example: Cogito

v What are costs/benefits of purchasing Cogito technology?u 5-minute screening test for each traveler.u Follow up inspection for those who fail screening.u $200K per machine.u Catch terrorists.

v How accurate is Cogito? How accurate do they want to be?

v If someone does not pass the screening, how likely is he/she to be a terrorist?

Cogito

v If someone does not pass the screening, how likely are they to be a terrorist?

v “The company's goal is to prove it can catch at least 90% of potential saboteurs -- a 10% false-negative rate -- while inconveniencing just 4% of innocent travelers.”

v Need assumptions.u 750,000,000 passengers fly per year (U.S.).u Assume 100 terrorists fly within a year.

The Probability Tree

96%

test positive

test positive

test negative

test negative

95.9999872%

0.0000120%

0.0000013%

0.00001333%

is a terrorist

is innocent

99.9998667%

4%

90%10%

3.9999995%

Pr( is a terrorist | test positive ) = 0. 0003% (about 1 chance in 333,000)

number of innocents hassled / year 30,000,000

number of terrorists who still get through 10

Independence of events

v Events A and B are independent if learning that B has happened does not change our assessed likelihood of A happening.

v Equivalent definitions:

u P(A | B) = P(A)u P(A | not B) = P(A)u P(B | A) = P(B)u P(B | not A) = P(B)

u P(A and B) = P(A B) = P(A) P(B)