The Dollar and Dime Game

Most mornings I take a walk to keep myself in shape. Usually I listen to an audiobook while walking, but for some reason my earbuds didn’t recharge properly overnight and this morning they didn’t work right. So, I had to take my constitutional in silence.

It so happens that yesterday I was revising Appendix H (The Binomial Probability Distribution) for the 6th edition of Intermediate Physics for Medicine and Biology. (Yes, you’re right, Gene Surdutovich and I are getting close to being done if we’re already up to the appendices.) As I was reviewing the material, I thought “it sure would be nice to have some more nontrivial but not too complicated word problems for this appendix.” So, as I hiked I came up with this:

Appendix H

Problem 6. You are a young college student who wants to make a little extra cash for living expenses. You also are an occasional Dungeons and Dragons player, so you have a twenty-sided die in the top drawer of your desk. You decide to set up what you call the “Dollar and Dime” game. Any student in your dormitory can come to you and pay you a dollar and a dime, and you will take out your twenty-sided die and roll it once. If it gives a one, you hand the student a crisp, new twenty dollar bill. If it it rolls a two through twenty, the student walks away empty handed. You’re pretty happy with the game. On average, the dollars earned cover the required payouts, and the dimes are all profit. The game becomes popular among your dormmates, and people stop by to play dozens of times each day.

A page from the Rodgers and
Hammerstein Song Book.

John comes to you late one Friday afternoon. He has invited Jane to attend the school musical Oklahoma! with him that evening (Jame loves musicals, especially those by Rodgers and Hammerstein), but two tickets will cost him $40, and all he has is $11. It’s too late to find a part-time job or to beg funds from his parents. His only chance to avoid reneging on the theater date is to get the needed cash by playing the Dollar and Dime game. John slaps the eleven bucks down on your desk and says “I wanna play ten times.”

Your first thought is to tell John to go to the bank and exchange the ten dollar bill for ten ones and the one dollar bill for ten dimes, so he can play the game properly. But John is on the school wrestling team, is six foot three, and weighs 270 pounds, so you decide to waive this technicality. You accept his $11, get out the twenty-sided die, and start rolling.

Ordinarily when playing this game you relax, knowing that in the long run you will make a profit. However, today you’re a bit nervous because you only have three portraits on Andrew Jackson in the envelope where you store the cash for your game. Earlier in the day, you told your wealthy roommate Peter about your situation, hoping he could cover you if needed (he declined). Now, if John wins the game four or more times, he’s gonna to be upset that you can’t pay him what you owe him, and John is not the kind of guy you want to make mad.

(a) What is the probability that John wins enough money to take Jane to Oklahoma!?

(b) What is the probability that you get clobbered by John?

(c) How do all these results change if Peter (who is annoyed that you converted your dorm room to a casino with people coming and going and noisily rolling that silly icosahedron at all hours of the night) loans John an extra $22, interest free?

The first step in solving this problem is to realize that this, indeed, can be modeled using the binomial probability distribution. In the 5th edition of IPMB, Russ Hobbie and I wrote

Consider an experiment with two mutually exclusive outcomes, which is repeated N times, with each repetition being independent of every other one. One of the outcomes is labeled “success”, the other is called “failure.”

The binomial distribution is given by Eq. H.2,

where N is the number of tries (John has $11 so he can play the game ten times, N = 10), p is the probability of success for each try (it is a twenty sided die, so p = 0.05), and n is the number of successes (rolling a one). John will make 20n dollars by playing the Dollar and Dime game. The key question is, what’s the probability P that John gets n wins.

The odds of John never rolling a one and leaving broke is

Yikes! He has 3:2 odds of losing everything. Next, the probability that John wins only once are 

Only one win will make John twenty bucks, so after paying $11 to play he’ll be nine dollars ahead, but that still isn’t enough to take Jane to see Curly give Laurey that ride in his surrey which, as you will recall, costs $40. He needs at least two wins for that. We now have enough information to answer part (a). The probability that John takes Jane to the show is one minus the probability that he doesn’t earn at least $40. So, John can avoid an unpleasant call to Jane (or, worse, escape being a no show) with a probability of 1 – 0.599 – 0.315 = 0.086. That means the odds are about 11:1 against making Jane happy. Looks like John’s in trouble.

John’s best chance is to win the Dollar and Dime game twice and earn the $40 needed for tickets. The odds are

Boy, that would be great. But if John is really lucky, he’ll win enough for the tickets plus some extra cash for a large popcorn and two medium soft drinks (which costs $18.49).  

There’s only a one percent chance of Jane getting her popcorn.

But wait. If John wins four or more times, you won’t have the cash to cover his winnings. Either he’ll thrash you, or (more likely) you’ll be forced to make a deal where you pay John all that you have, $60, and promise to return his original investment of $11, and grovel before him begging for mercy. That would be good news for John. He would walk away with at least $71, and perhaps more if he knows how to drive a hard bargain (after all, you don’t want to end up daid, like poor Jud). What are the odds he’ll bust the bank? 

We should also add in the chance that John will win five times, or six, or more, but those will be very small (calculate them yourself if you don’t believe me). So, the probability of a disaster (for you, not for John) is about one part per thousand, or a tenth of a percent. The odds are small, but the consequences would be dire (with you possibly ending up in the hospital), so you’re still nervous until John finishes all ten of his rolls.

Now, consider the final twist to the story. Imagine that when your so-called “friend” Peter sees John arrive, he pulls him aside, gives him a wink, and loans him another $22. (Pete could have easily just lent $29 so John would have enough to cover the cost of his date with Jane, but that would defeat his purpose, wouldn’t it?). Now John has $33 to spend on the Dollar and Dime game. The only thing that changes is N increases from ten to thirty. How does that change the probabilities? You can work out the details. I’ll just state the results.

n        P 

  0     0.215 

  1     0.339 

  2     0.259 

  3     0.127 

 4     0.045

The chances of John taking Jane to the musical is now 0.446, so the odds are approaching 50-50. Still not great odds, but much better than before. John’s starting to dream that after he takes Jane to Oklahoma! “people will say we’re in love.” More importantly for you (and for that evil Peter), the odds of busting the bank are now 6%. So, at no cost to himself, Peter just increased the odds of shutting down the hated casino by a factor of sixty. Win or lose, you vow to start looking for another roommate; one who doesn’t know as much math.

By the time I came up with this homework problem, I had just about finished my walk. The problem has no biology or medicine in it, so it probably won’t make it into the revised sixth edition. With any luck, tomorrow I’ll be back to the audio book (and, oh, what a beautiful morning that will be). By the way, our goal is to submit the 6th edition of IPMB to our publisher, Springer, before the end of the year. It’s gonna be close, but we just might make it.

Oklahoma!

https://www.youtube.com/watch?v=ZbrnXl2gO_k&list=RDZbrnXl2gO_k

 

 Oh, What a Beautiful Morning

https://www.youtube.com/watch?v=O5APc0z49wg&list=RDO5APc0z49wg

 

People Will Say We're in Love

 

https://www.youtube.com/watch?v=VEwVAV3VPw4

The Surrey with the Fringe on Top

https://www.youtube.com/watch?v=BIG_GVE-KiE&list=RDBIG_GVE-KiE

Kansas City

https://www.youtube.com/watch?v=M6pmZE1Qtyw&list=RDBIG_GVE-KiE

 

The Farmer and the Cowman

https://www.youtube.com/watch?v=G85aEsgMDwA&list=RDG85aEsgMDwA 

I Cain't Say No

https://www.youtube.com/watch?v=aExjjv2Klrs&list=RDaExjjv2Klrs&start_radio=1

 

Poor Jud is Daid

https://www.youtube.com/watch?v=Euq8Z4l6Iwk&t=171s

Maxwell's Spot

One of my scientific heroes is James Clerk Maxwell. Maxwell (1831–1879) was a Scottish physicist known for developing Maxwell’s equations of electricity and magnetism, and for his work on statistical mechanics. But Maxwell also studied the eye and was an early researcher of color vision.

One of Maxwell’s interesting but not staggeringly important discoveries was Maxwell’s Spot. If for no other reason that I am a big Maxwell fan, today I want to examine his spot. I’ll base my discussion on a 2003 article titled “The Differential Contribution of Macular Pigments and Foveal Anatomy to the Perception of Maxwell’s Spot and Haidinger’s Brushes” by Gary Misson, Rebekka Heitmar, Richard Armstrong, and Stephen Anderson in the journal Vision. I will quote from it with references removed. The article begins

Normally sighted individuals can perceive a short-lived darkened spot at the point of fixation while viewing a plain white surface through a dichroic filter transmitting a mixture of long- and short-wavelength lights. This entoptic phenomenon, known as Maxwell’s spot (MS), was first described in detail by James Clerk Maxwell in 1856.

One of my goals today will be to merely explain what unfamiliar words mean. First, what is a “dichroic filter”? A dichroic filter uses interference rather than absorption to filter light. Interference was discussed only briefly in Intermediate Physics for Medicine and Biology, when Russ Hobbie and I described optical coherence tomography. A dichroic filter is typically made up of many thin layers, each which can reflect light. It creates colors in a similar way that a thin film of oil on the surface of water reflects some colors and not other. It depends on if the reflected or transmitted light from different layers interfere constructively or destructively. One advantage of a dichroic filter is that it can be very selective about what light is transmitted.

Next is “entoptic.” An entoptic phenomenon is a visual effect that arises from a source or structure within the eye itself. Examples of vision phenomena that are NOT entoptic include hallucinations (arising in the brain) and mirages (arising in the optical refraction of light in the environment). Entoptic phenomena would include floaters arising from the shadow of tiny objects in the vitreous humor of the eye, and phosphenes arising from mechanical or electrical excitation of the retina. Whatever Maxwell’s Spot is, it happens because of something within the eye itself. Misson et al. continue

The most widely accepted hypothesis proposed for the origin of the peripheral zones in [Maxwell’s Spot], and its documented perceptual variations, is absorption of blue light by macular pigments that result in a reduction of foveal photoreceptor illumination.

First, what is the “macula?” It is an oval-shaped region in the center of the retina with a diameter of about 5 mm where there is a high density of cone cells responsible for color vision. At the center of the macula is a region of about 1.5 mm diameter that has the highest density of cone cells called the fovea.

To fully understand the vision, you need to realize that humans have what’s called an inverted retina. That is, the light-sensing rods and cones are at the back of the retina, behind the retina’s neurons and capillaries, so light must pass through these other structures before reaching the light-sensing cells. You may ask, why does the retina have this seemingly backwards structure? I’ll tell you. I don’t know. But it does.

The macula also contains pigments that absorb light. Like the neurons and capillaries, these pigments (at least some of them) are located in front of the rods and cones. Pigments are molecules that absorb certain wavelengths of light. Two of the main macular pigments are called lutein and zeaxanthin. These are carotenoids, which are the pigments that give color to pumpkins, carrots, and daffodils. In general, carotenoids absorb blue and violet light. So, the reason a carrot is orange is that when white light shines on it the caretenoids absorb much of the blue light, so the light reflected by the carrot (which is the light you see) is mainly the red and orange light that was not absorbed. This is also why the macula itself looks yellow when viewed with a ophthalmoscope. 

Figure 3 from Misson et al. (2003). The image was obtained
using optical coherence tomography. Light comes in from above.
The bright areas on the top left and right are the macular
pigments. There is also a lot of pigment below the retina,
but that does play a key role in producing Maxwell’s Spot.

So, now we get to the cause of Maxwell’s Spot. Misson and his coauthors write

The results of this study support the theory that the principal mechanism of [Maxwell’s Spot] generation is pre-receptoral screening by macular pigment.

In other words, the pigments in front of the macula (through which the incoming light must pass to reach the rods and cones) filters out some of the blue light. As white light enters the eye, the macular pigments remove some of the blue, but the rest of the retina which does not have these pigments lets all the light through. So, a white screen appears white except at the center spot right where the eye is fixated on with highest spatial resolution and best color vision, and that spot is darker and redish. That, dark red region is Maxwell’s Spot. Note that this phenomenon arises because of the distribution of pigments within the eye; it is entoptic. Because everybody’s pigment distribution and macula arrangement can vary, so can everyone’s perception of Maxwell’s Spot.

Maxwell’s Spot was not Maxwell’s only contribution to vision physiology. He was one of the founders of the theory of trichromatic color vision, which states that there are three types of cone cells in the retina (red, green, and blue) that are responsible for our perception of color. There is no telling how much more Maxwell might have contributed to both physics and physiology if he had not died of cancer at the tragically young age of 48.

Why Are There So Few Aerial Plankton?

Air and Water,
by Mark Denny.

In his book Air and Water, Mark Denny asks an oddball question: Why are there so few aerial plankton? In my mind, this question transforms into: Why are there no flying blue whales, sucking in mouthfuls of air and filtering out tiny organisms for food? Here is what Denny says:

A general characteristic of aquatic (especially marine) environments is the presence of planktonic life. A cubic meter of water taken from virtually anywhere in a stream, lake, or ocean is teeming with small, suspended organisms. In fact, the concentration of these plants and animals is such that many kinds of invertebrates, including clams, mussels, anemones, polychaete worms, and bryozoans, can reliably use planktonic particles as their sole source of food. In contrast, air is relatively devoid of suspended matter. A cubic meter of air might contain a few bacteria, a pollen grain or two, and very occasionally a flying insect or wind-borne seed. Air is so depauperate compared to the aquatic ‘soup’ that few terrestrial animals manage to make a living by straining their food from the surrounding fluid. Web-building spiders are the only example that comes to mind.

To understand why, my first inclination is to examine the balance between gravity, thermal motion, and concentration. You can use a Boltzmann factor, e^–mgh/k_BT, to determine how the concentration changes with height h, assuming particles of mass m are in contact with a fluid at temperature T (g is the acceleration of gravity and k_B is Boltzmann’s constant). But there’s one problem: the Peclet number is often large, meaning advection dominates diffusion. In other words, air or water currents are more effective than diffusion for mixing (like in a blender). In many cases the problem is even worse: the flow is turbulent, which tends to mix materials much more rapidly than diffusion does. In some ways the analysis of turbulent mixing is similar to the case of diffusion. The flux of particles is proportional to the concentration gradient, but the constant of proportionality is not the diffusion constant but instead the turbulent diffusivity. Denny does the analysis in more detail than I can go into here (turbulent flow is always complex). But he states his conclusion clearly and simply

The sinking rates of particles in air are just too high to allow them to remain passively suspended and as a result, aerial plankton are sparse. In water, slow sinking speeds insure that many particles are suspended, and the plankton is plentiful. The abundance of aquatic suspension feeders and the scarcity of terrestrial ones, can therefore be thought of as a direct consequence of the differences in density and viscosity between air and water.

One other factor plays a role here: buoyancy. If the small organisms have a density approximately that of water, then tiny aquatic animals would be almost neutrally buoyant, so they’d be easy to suspend. In air, however, buoyancy plays almost no roll, so these little animals “seem” to be much denser.

It looks like I should abandon my search for a giant flying suspension feeder, resembling a blimp with a big mouth to suck in large amounts of air that it filters to extract food. Too bad. I was looking forward to befriending one, if the physics had only allowed it.

 https://www.youtube.com/watch?v=NivDXM88oCo

Does Stokes’ Law Hold for a Bubble?

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Stokes’ law. When a small sphere, of radius a, moves with speed U through a fluid having a viscosity η, the drag force D is 6πηaU. This result is well known, but does it apply to a gas bubble moving in water?

Life in Moving Fluids,
by Steven Vogel.

I was reading through Life in Moving Fluids, by Steven Vogel, when I came across the answer. Vogel considers a fluid sphere moving in a fluid medium. His Eq. 15.8 is

Here, η_ext is the viscosity of the external fluid and η_int is the viscosity of the internal fluid.

Suppose you have a sphere of water in air (say, a raindrop falling from the sky toward earth). Then η_int = η_water = 10^–3 Pa s and η_ext = η_air = 2 × 10^–5 Pa s. Thus η_ext/η_int = 0.02. For our purposes, this is nearly zero, and the drag force reduces to Stokes’ law, D = 6πη_extaU.

Now, consider a sphere of air in water (say, a bubble rising toward the surface of a lake). Then η_int = η_air = 2 × 10^–5 Pa s and η_ext = η_water = 10^–3 Pa s. Thus η_ext/η_int = 50. For our purposes, this is nearly infinity, and the drag force becomes D = 4πη_extaU. Yikes! Stokes’ law does not hold for a bubble. Who knew? (Vogel knew.)

Apparently when the sphere is a fluid, internal motion occurs, as shown in Vogel’s picture below.

 

Note that at the edge of the sphere, the internal and external flows are in the same direction. This changes the boundary condition at the surface. A rigid sphere would obey the no-slip condition, but a fluid sphere does not because the internal fluid is moving.

Although Vogel doesn’t address this, I wonder what the drag force is on a sphere of water in water? Does this even make sense? Perhaps we would be better off considering a droplet of some liquid that has the same viscosity as water moving through water (I can imaging this might happen in a microfluidics apparatus). In that case the drag force becomes D = 5πη_extaU. I must confess, I’m not sure if the derivation of the general equation is valid in this case, but I don’t see why it shouldn’t be.

There are all kinds of little jewels inside Vogel’s book. I sure wish he were still around.