Vanity Fair bungles probability example

There's an interesting article about Danny Kahneman in Vanity Fair, written by Michael Lewis. Kahneman's book Thinking: Fast And Slow is an interesting review of the state of decision psychology and well worth reading, as it the Vanity Fair article.

But the quiz attached to that article is an example of how not to popularize technical content.

This example, question 2, is wrong:

A team of psychologists performed personality tests on 100 professionals, of which 30 were engineers and 70 were lawyers. Brief descriptions were written for each subject. The following is a sample of one of the resulting descriptions:


Jack is a 45-year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies, which include home carpentry, sailing, and mathematics. 

What is the probability that Jack is one of the 30 engineers?


A. 10–40 percent
B. 40–60 percent
C. 60–80 percent
D. 80–100 percent


If you answered anything but A (the correct response being precisely 30 percent), you have fallen victim to the representativeness heuristic again, despite having just read about it. 

No. Most people have knowledge beyond what is in the description; so, starting from the appropriate prior probabilities, $p(law) = 0.7$ and $p(eng) = 0.3$, they update them with the fact that engineers like math more than lawyers, $p(math|eng) >> p(math|law)$. For illustration consider

$p(math|eng) = 0.5$; half the engineers have math as a hobby.
$p(math|law) = 0.001$; one in a thousand lawyers has math as a hobby.

Then the posterior probabilities (once the description is known) are given by

$p(eng|math) = \frac{ p(math|eng) \times p(eng)}{p(math)}$

$p(law|math) = \frac{ p(math|law) \times p(law)}{p(math)}$

with $p(math) = p(math|eng) \times p(eng) + p(math|law) \times p(law)$. In other words, with the conditional probabilities above,

$p(eng|math) = 0.995$
$p(law|math) = 0.005$

Note that even if engineers as a rule don't like math, only a small minority does, the probability is still much higher than 0.30 as long as the minority of engineers is larger than the minority of lawyers*:

$p(math|eng) = 0.25$ implies $p(eng|math) = 0.991$
$p(math|eng) = 0.10$ implies $p(eng|math) = 0.977$
$p(math|eng) = 0.05$ implies $p(eng|math) = 0.955$
$p(math|eng) = 0.01$ implies $p(eng|math) = 0.811$
$p(math|eng) = 0.005$ implies $p(eng|math) = 0.682$
$p(math|eng) = 0.002$ implies $p(eng|math) = 0.462$

Yes, that last case is a two-to-one ratio of engineers who like math to lawyers who like math; and it still falls out of the 10-40pct category.

I understand the representativeness heuristic, which mistakes $p(math|eng)/p(math|law)$ for $p(eng|math)/p(law|math)$, ignoring the base rates, but there's no reason to give up the inference process if some data in the description is actually informative.

-- -- -- --
* This example shows the elucidative power of working through some numbers. One might be tempted to say "ok, there's some updating, but it will probably still fall under the 10-40pct category" or "you may get large numbers with a disproportionate example like one-half of the engineers and one-in-a-thousand lawyers, but that's just an extreme case." Once we get some numbers down, these two arguments fail miserably.

Numbers are like examples, personas, and prototypes: they force assumptions and definitions out in the open.

Less

I found a magic word and it's "less."

On September 27, 2011, I decided to run a lifestyle experiment. Nothing radical, just a month of no non-essential purchases, the month of October 2011. These are the lessons from that experiment.


Separate need, want, and like

One of the clearest distinctions a "no non-essential purchases" experiment required me to make was the split between essential and non-essential.

Things like food, rent, utilities, gym membership, Audible, and Netflix I categorized as essential, or needs. The first three for obvious reasons, the last three because the hassle of suspending them wasn't worth the savings.

A second category of purchases under consideration was wants, things that I felt that I needed but could postpone the purchase until the end of the month. This included things like Steve Jobs's biography, for example. I just collected these in the Amazon wish list.

A third category was likes. Likes were things that I wanted to have but knew that I could easily live without them. (Jobs's biography doesn't fall into this category, as anyone who wants to discuss the new economy seriously has to read it. It's a requirement of my work, as far as I am concerned.) I placed these in the Amazon wish list as well.

Over time, some things that I perceived as needs were revealed as simply wants or even likes. And many wants ended up as likes. This means that just by delaying the decision to purchase for some time I made better decisions.

This doesn't mean that I won't buy something because I like it (I do have a large collection of music, art, photography, history, science, and science fiction books, all of which are not strictly necessary). What it means is that the decision to buy something is moderated by the preliminary categorization into these three levels of priority.

A corollary of this distinction is that it allows me to focus on what is really important in the activities that I engage in. I summarized some results in the following table (click for bigger):



One of the regularities of this table is that the entries in the middle column (things that are wrongly emphasized) tend to be things that are bought, while entries in the last column (what really matters) tend to be things that are learned or experienced.


Correct accounting focusses on time, not on nominal money

Ok, so I can figure out a way to spend less in things that are not that necessary. Why is this a source of happiness?

Because money to spend costs time and I don't even get all the money.

When I spend one hour working a challenging technical marketing problem for my own enjoyment, I get the full benefit of that one hour of work, in the happiness solving a puzzle always brings me. When I work for one hour on something that I'd rather not be doing for a payment of X dollars, I get to keep about half of those X dollars (when everything is accounted for). I wrote an illustration of this some time ago.

In essence, money to spend comes, at least partially from doing things you'd rather not do, or doing them at times when you'd rather be doing something else, or doing them at locations that you'd rather not travel to. I like the teaching and research parts of my job, but there are many other parts that I do because it's the job. I'm lucky in that I like my job; but even so I don't like all the activities it involves.

The less money I need, the fewer additional things I have to do for money. And, interestingly, the higher my price for doing those things. (If my marginal utility of money is lower, you need to pay more for me to incur the disutility of teaching that 6-9AM on-location exec-ed seminar than you'd have to pay to a alternate version of me that really wants money to buy the latest glued "designer" suit.)


Clarity of purpose, not simply frugality, is the key aspect

I'm actually quite frugal, having never acquired the costly luxury items of a wife and children, but the lessons here are not about frugality, rather about clarity of purpose.

I have a $\$$2000 17mm ultra-wide angle tilt-shift lens on my wishlist, as a want. I do want to buy it, though I don't need it for now. Once I'm convinced that the lens on the camera, rather than my skills as a photographer, is the binding constraint in my photography, I plan to buy the lens. (Given the low speed at which my photography skill is improving, this may be a non-issue. ☺)

Many of our decisions are driven by underlying identity or symbolic reasons; other decisions are driven by narrowly framed problems; some decisions are just herd behavior or influenced by information cascades that overwhelm reasonable criteria; others still are purely hedonic, in-the-moment, impulses. Clarity of purpose avoids all these. I ask:

Why am I doing this, really?

I was surprised at how many times the answer was "erm...I don't know," "isn't everybody?" or infinitely worse "to impress X." These were not reasonable criteria for a decision. (Note that this is not just about purchase decisions, it's about all sorts of little decisions one makes every day, which deplete our wallets but also our energy, time, and patience.)

Clarity of purpose is hard to achieve during normal working hours, shopping, or the multiple activities that constitute a lifestyle. Borrowing some tools designed for lifestyle marketing, I have a simple way to do a "personal lifestyle review" using the real person "me" as the persona used in lifestyle marketing analysis. Adapted from the theory, it is:

1. Create a comprehensive list of stuff (not just material possessions, but relationships, work that is pending, even persons in one's life).

2. Associate the each entry in the stuff to a sub-persona (for non-marketers this means to a part of the lifestyle that is more or less independent of the others).

3. For each sub-persona, determine the activities which have given origin to the stuff.

4. Evaluate the activities using the "clarity of purpose" criterion: why am I doing this?

5. Purge the activities that are purely symbolic and those that were adopted for hedonic reasons but do not provide the hedonic rewards associated with their cost (in money, constraints to life, time, etc), plus any functional activities that are no longer operative.

6. Guide life decisions by the activities that survive the purge. Revise criteria only by undergoing a lifestyle review process, not by spur-of-the-moment impulses.

(This procedure is offered with no guarantees whatsoever; marketers may recognize the underlying structure from lifestyle marketing frameworks with all the consumer decisions reversed.)


Less. It works for me.


A final, cautionary thought: if the ideas I wrote here were widely adopted, most economies would crash. But I don't think there's any serious risk of that.

Thinking skill, subject matter expertise, and information

Good thinking depends on all three, but they have different natures.

To illustrate, I'm going to use a forecasting tool called Scenario Planning to determine my chances of dating Milla Jovovich.

First we must figure out the causal structure of the scenario. The desired event, "Milla and I live happily ever after," we denote by $M$. Using my subject matter expertise on human relationships, I postulate that $M$  depends on a conjunction of two events:

• Event $P$  is "Paul Anderson – her husband – runs away with a starlet from one of his movies"
• Event $H$  is "I pick up the pieces of Milla's broken heart"

So the scenario can be described by $P \wedge H \Rightarrow M$. And probabilistically,

$\Pr(M) = \Pr(P) \times \Pr(H).$

Now we can use information from the philandering of movie directors and the knight-in-shining-armor behavior of engineering/business geeks [in Fantasyland, where Milla and I move in the same circles] to estimate $\Pr(P) =0.2$  (those movie directors are scoundrels) and $\Pr(H)=0.1$  (there are other chivalrous nerds willing to help Milla) for a final result of $\Pr(M)=0.02$, or 2% chance.

Of course, scenario planning allows for more granularity and for sensitivity analysis.

We could decompose event $P$  further into a conjunction of two events, $S$  for "attractive starlet in Paul's movies" and $I$  for "Paul goes insane and chooses starlet over Milla." We could now determine $\Pr(P)$  from these events instead of estimating it directly at 0.2 from the marital unreliability of movie directors in general, using $\Pr(P) = \Pr(S) \times \Pr(I).$

Or, going in another direction, we could do a sensitivity analysis. Instead of assuming a single value for $\Pr(P)$ and $\Pr(H)$, we could find upper and lower bounds, say $0.1 < \Pr(P) < 0.3$  and $0.05 < \Pr(H) < 0.15$. This would mean that  $0.005 < \Pr(M) < 0.045$.

Of course, if instead of the above interpretation we had

• Event $P$  is "contraction in the supply of carbon fiber"
• Event $H$  is "increase in the demand for lightweight camera tripods and monopods"
• Event $M$  is "precipitous increase in price and shortages of carbon fiber tennis rackets"

the same scenario planning would be used for logistics management of a sports retailer provisioning.

Which brings us to the three different competencies needed for scenario planning, and more generally, for thinking about something:

Thinking skill is, in this case, knowing how to use scenarios for planning. It includes knowing that the tool exists, knowing what its strengths and weaknesses are, how to compute the final probabilities, how to do sensitivity analysis, and other procedural matters. All the computations above, which don't depend on what the events mean are pure thinking skill.

Subject matter expertise is where the specific elements of the scenario and their chains of causality come from. It includes knowing what to include and what to ignore, understanding how the various events in a specific subject area are related, and understanding the meaning of the events (as opposed to just computing inferential chains like an algorithm). So knowing that movie directors tend to abandon their wives for starlets allows me to decompose the event $P$  into $S$  and $I$  in the Milla example. But only an expert in the carbon fiber market would know how to decompose $P$  when it becomes the event "contraction in the supply of carbon fiber."

Information, in this case, are the probabilities used as inputs for calculation, as long as those probabilities come from data. (Some of these, of course, could be parameters of the scenario, which would make them subject matter expertise. Also, instead of a strict implication we could have probabilistic causality.) For example, the $\Pr(P)=0.2$  could be a simple statistical count of how many directors married to fashion models leave their wives for movie starlets.

Of these three competencies, thinking skill is the most transferrable: knowing how to do the computations associated with scenario planning allows one to do them in military forecasting or in choice of dessert for dinner. It is also one that should be carefully learned and practiced in management programs but typically is not given the importance its real-world usefulness would imply.

Subject matter expertise is the hardest to acquire – and the most valuable – since it requires both acquiring knowledge and developing judgment. It is also very hard to transfer: understanding the reactions of retailers in a given area doesn't transfer easily to forecasting nuclear proliferation. 

Information is problem-specific and though it may cost money it doesn't require either training (like thinking skill) or real learning (like subject matter expertise). Knowing which information to get requires both thinking skill and subject matter expertise, of course.

Getting these three competencies confused leads to hilarious (or tragic) choices of decision-maker. For example, the idea that "smart is what matters" in recruiting for specific tasks ignores the importance of subject matter expertise.*

Conversely, sometimes a real subject matter expert makes a fool of himself when he tries to opine on matters beyond his expertise, even ones that are simple. That's because he may be very successful in his area due to the expertise making up for faulty reasoning skills, but in areas where he's not an expert those faults in reasoning skill become apparent.

Let's not pillory a deceased equine by pointing out the folly of making decisions without information; on the other hand, it's important to note the idiocy of mistaking someone who is well-informed (and just that) for a clear thinker or a knowledgeable expert.

Understanding the structure of good decisions requires separating these three competencies. It's a pity so few people do.

-- -- -- --

* "Smart" is usually a misnomer: people identified as "smart" tend to be good thinkers, not necessarily those who score highly on intelligence tests. Think of intelligence as raw strength and thinking as olympic weightlifting: the first helps the second, but strength without skill is irrelevant. In fact, some intelligent people end up being poor thinkers because they use their intelligence to defend points of view that they adopted without thinking and turned out to be seriously flawed.

Note 1: This post was inspired by a discussion about thinking and forecasting with a real clear thinker and also a subject matter expert on thinking, Wharton professor Barbara Mellers.

Note 2: No, I don't believe I have a 2% chance of dating Milla Jovovich. I chose that example precisely because it's so far from reality that it will give a smile to any of my friends or students reading this.

The costly consequences of misunderstanding cost

Apparently there's growing scarcity of some important medicines. And why wouldn't there be?

Some of these medicines are off-patent, some are price-controlled (at least in most of the world), some are bought at "negotiated" prices where one of the parties negotiating (the government) has the power to expropriate the patent from the producer. In other words, their prices are usually set at variable cost plus a small markup.

Hey, says Reggie the regulator, they're making a profit on each pill, so they should produce it anyway.

(Did you spot the error?)

(Wait for it...)

(Got it yet?)

Dear Reggie: pills are made in these things called "laboratories," that are really factories. Factories, you may be interested to know, have something called "capacity constraints," which means that using a production line for making one type of pill precludes that production line from making a different kind of pill. Manufacturers are in luck, though, because most production lines can be repurposed from one medication to another with relatively small configuration cost.

Companies make their decisions based on opportunity costs, not just variable costs. If they have a margin of say 90 cents/pill for growing longer eyelashes (I'm not kidding, there's a "medication" for that) and say 5 cents/pill to cure TB, they are going to dedicate as much of their production capacity to the eyelash-elongating "medication" as they can.* (They won't stop making the TB medication altogether because that would be bad for public relations.)

Funny how these things work, huh?

-----------
* Unless they can make more than eighteen times more TB pills than eyelash "medicine" pills with the same production facilities, of course.

Books on teaching and presentations

During a decluttering of my place, I had to make decisions about which books to keep; these are some that I found useful for teaching and presentations, and I'm therefore keeping:

They are stacked by book size (for stability), but I'll group them in four major topics: general presentation planning and design; teaching; speechwriting; and visuals design.

1. Presentation planning and design

Edward Tufte's Beautiful Evidence is not just about making presentations, rather it's about analyzing, presenting, and consuming evidence.

Lani Arredondo's How to Present Like a Pro is the only "general presentation" book I'm keeping (and I'm still pondering that, as most of what it says is captured in my 3500-word post on preparing presentations). It's not especially good (or bad), it's just the best of the "general presentation" books I have, and there's no need for more than one. Whether I need one given Beautiful Evidence is an open question.

Donald Norman's Living With Complexity and Things That Make Us Smart are not about presentations, rather about designing cognitive artifacts (of which presentations and teaching exercises are examples) for handling complex and new units of knowledge.

Chip and Dan Heath's Made to Stick is a good book on memorability; inasmuch as we expect our students and audiences to take something away from a speech, class, or exec-ed, making memorable cognitive artifacts is an important skill to have.

Steve Krug's Don't Make Me Think is about making the process of interactions with cognitive artifacts as simple as possible (the book is mostly about the web, but the principles therein apply to presentation design as well).

Alan Cooper's The Inmates Are Running The Asylum is similar to Living With Complexity, with the added benefit of explicitly addressing the use of personas for designing complex products (a very useful product design tool for classes, I think).

I had other books on the general topic of presentations that I am donating/recycling. Most of them spend a lot of space discussing the management of stage fright, a problem with which I am not afflicted.

If I had to pick just one to keep, I'd choose Beautiful Evidence. (The others, except How To Present Like a Pro, are research-related, so I'd keep them anyway.)


2. Teaching

As I've mentioned previously, preparing instruction is different from preparing presentations. The two books I recommended then are the two books I'm keeping:

Tools for teaching, by Barbara Gross Davis covers every element of course design, class design, class management, and evaluation. It is rather focussed on institutional learning (like university courses), but many of the issues, techniques, and checklists are applicable in other instruction environments.

Designing effective instruction, by Gary Morrison, Steven Ross, and Jerrold Kemp, complements Tools for teaching. While Tools for Teaching has the underlying model of a course, this book tackles the issues of training and instruction from a professional service point of view. (In short: TfT is geared towards university classes, DEI is geared towards firm-specific Exec-Ed.)

I had other books on the general topic of teaching (and a number of books on academic life) that I am donating/recycling.


3. Speechwriting and public speaking

Speak like Churchill, stand like Lincoln, by James Humes, should be mandatory reading for anyone who ever has to make a public speech. Of any kind. Humes is a speechwriter and public speaker by profession and his book gives out practical advice on both the writing and the delivery. I have read many books on public speaking and this one is in a class of its own.

I have a few books from the Toastmasters series; I'm keeping (for now at least) Writing Great Speeches and Choosing Powerful Words, though their content overlaps a lot with Virginia Tufte's Beautiful Sentences, a book I'm definitely keeping as part of my writing set.

I'm probably keeping Richard Dowis's The Lost Art of The Great Speech as a good reference for styles and as motivation reading. (Every so often one needs to be reminded of why one does these things.)

I have other books on writing, in general, but the ones in the pile above are specific to speechwriting. I'm throwing out a few books on the business of speechwriting; they are so bad that I thought of keeping them as satire. Donating them would be an act of cruelty towards the recipients.

If I had to pick just one book on speechwriting, I'd go with Speak like Churchill, Stand like Lincoln. Hands down the best in the category, and I've read many.

4. Visuals design

Yes, the design of visuals for presentations or teaching, not Visual Design the discipline.

Edward Tufte's books are the alpha and the omega in this category. Anyone with any interest in information design should read these books carefully and reread them often.

The Non-Designer Design Book, by Robin Williams lets us in on the secrets behind what works visually and what doesn't. It really makes one appreciate the importance of what appears at first to be over-fussy unimportant details. I complement this with The Non-Designer Type Book and Robin Williams Design Workshop, the first specifically for type, the second as an elaboration of the Non-Designer Design Book.

Universal principles of design, by William Lidwell, Kristina Holden, and Jill Butler is a quick reference for design issues. I also like to peruse it regularly to get some reminders of design principles. It's organized alphabetically and each principle has a page or two, with examples.

Perhaps I'm a bit focussed on typography (a common symptom of reading design books, I'm told), but Robert Bringhurst's The Elements of Typographic Style is a really good and deeply interesting book on the subject. Much more technical than The Non-Designer Type Book, obviously, and the reason why I hesitate to switch from Adobe CS to iWork for my handouts.

Zakia and Page's Photographic Composition: A visual guide is very useful as a guide to laying out materials for impact. Designing the visual flow of a slide (or a handout) -- when there are options, of course, this is not about "reshaping" statistical charts -- helps tell a story even without narration or animation.

I had some other books on the general topic of slide design, which I am donating. I also have a collection of books on art, photography, and design in general, which affords me a reference library. (That collection I'm keeping.)

If I had to pare down the set further, the last ones I'd give up are the four Tufte books. If forced to pick just one (in addition to Beautiful Evidence, which fills the presentation category above), I'd choose The Visual Display of Quantitative Information, because that's the most germane to the material I cover.


CODA: A smaller set

Not that I'm getting rid of the books in the larger set above (that's the set that I'm keeping), but I think there's a core set of books I should reread at least once a year. Unsurprisingly, those are the same books I'd pick if I really could have only one per category (or one set for the last category):

Note that the Norman, Heath Bros, Krug, Cooper books and my collection of art, photography, and design books are exempted from this choice, as they fall into separate categories: research-related or art. I also have several books on writing (some of them here).

And the books that didn't make the pile at the beginning of the post? Those, which I'm donating or recycling, make up a much larger pile (about 50% larger: 31 books on their way out).

Somewhat related posts:

Posts on presentations in my personal blog.

Posts on teaching in my personal blog.

Posts on presentations in this blog.

My 3500-word post on preparing presentations.

What to do about psychological biases? The answer tells a lot... about you.

There are many documented cases of behavior deviating from the normative "rational" prescription of decision sciences and economics. For example, in the book Predictably Irrational, Dan Ariely tells us how he got a large number of Sloan School MBA students to change their choices using an irrelevant alternative.

The Ariely example has two groups of students choose a subscription type for The Economist. The first group was given three options to choose from: (online only, $\$60$); (paper only, $\$120$); or (paper+online, $\$120$). Overwhelmingly they chose the last option. The second group was given two options : (online only, $\$60$) or (paper+online $\$120$). Overwhelmingly they chose the first option.

Since no one chooses the (paper only, $\$120$) option, it should be irrelevant to the choices. However, removing it makes a large number of respondents change their minds. This is what is called a behavioral bias: an actual behavior that deviates from "rational" choice. (Technically these choices violate the Strong Axiom of Revealed Preference.)

(If you're not convinced that the behavior described is irrational, consider the following isomorphic problem: a waiter offers a group of people three desserts: ice cream, chocolate mousse, and fruit salad; most people choose the fruit salad, no one chooses the mousse. Then the waiter apologizes: it turns out there's no mousse. At that point most of the people who had ordered fruit salad switch to ice cream. This behavior is the same -- use some letters to represent options to remove any doubt -- as the one in Ariely's example. And few people would consider the fruit salad to ice-cream switchers rational.)

Ok, so people do, in some cases (perhaps in a majority of cases) behave in "irrational" ways, as described by the decision science and economics models. This is not entirely surprising, as those models are abstractions of idealized behavior and people are concrete physical entities with limitations and -- some argue -- faulty software.

What is really enlightening is how people who know about this feel about the biases.

IGNORE. Many academic economists and others who use economics models try to ignore these biases. Inasmuch as these biases can be more or less important depending on the decision, the persons involved, and the context, this ignorance might work for the economists, for a while. However, pretending that reality is not real is not a good foundation for Science, or even life.

ATTACK. A number of people use the existence of biases as an attack on established economics. This is how science evolves, with theories being challenged by evidence and eventually changing to incorporate the new phenomena. Some people, however, may be motivated by personal animosity towards economics and decision sciences; this creates a bad environment for knowledge evolution -- it becomes a political game, never good news for Science.

EXPLOIT. Books like Nudge make this explicit, but many people think of these biases as a way to manipulate others' behavior. Manipulate is the appropriate verb here, since these people (maybe with what they think is the best of intentions -- I understand these pave the way to someplace...) want to change others' behavior without actually telling these others what they are doing. In addition to the underhandedness that, were this a commercial application, the Nudgers would be trying to outlaw, this type of attitude reeks of "I know better than others, but they are too stupid to agree." Underhanded manipulation presented as a virtue; the world certainly has changed a lot.

ADDRESS AND MANAGE. A more productive attitude is to design decisions and information systems to minimize the effect of these biases. For example, in the decision above, both scenarios could be presented, the inconsistency pointed out, and then a separate part-worth decision could be addressed (i.e. what are each of the two elements -- print and online -- worth separately?). Note that this is the one attitude that treats behavioral biases as damage and finds way to route decisions around them, unlike the other three attitudes.


In case it's not obvious, my attitude towards these biases is to address and manage them.

Probability interlude: from discrete events to continuous time

Lunchtime fun: the relationship between Bernoulli and Exponential distributions.

Let's say the probability of Joe getting a coupon for Pepsi in any given time interval $\Delta t$, say a month, is given by $p$. This probability depends on a number of things, such as intensity of couponing activity, quality of targeting, Joe not throwing away all junk mail, etc.

For a given integer number of months, $n$, we can easily compute the probability, $P$, of Joe getting at least one coupon during the period, which we'll call $t$, as

$P(n) = 1 - (1-p)^n$.

Since the period $t$  is $t= n \times \Delta t$, we can write that as

$P(t) = 1 - (1-p)^{\frac{t}{\Delta t}}.$

Or, with a bunch of assumptions that we'll assume away,

$P(t) = 1- \exp\left(t \times \frac{\log (1-p)}{\Delta t}\right).$

Note that $\log (1-p)<0$. Defining $r = - \log (1-p) /\Delta t$, we get

$P(t) = 1 - \exp (- r t)$.

And that is the relationship between the Bernoulli distribution and the Exponential distribution.

We can now build continuous-time analyses of couponing activity. Continuous analysis is much easier to do than discrete analysis. Also, though most simulators are, by computational necessity, discrete, building them based on continuous time models is usually simpler and easier to explain to managers using them.