How Likely are You to Win the TNT Million Dollar Challenge?

TNT is offering a million dollars to anyone who picks the top ten drivers – in order – at any of the six races they broadcast.  You have up until 25% of the race has been run to lock in your selections, which means up to mile 100 at Daytona this weekend.   How likely are you to win?

You have a 1 in 43 chance of picking the first driver correctly.  There are now 42 drivers left and you have a 1 in 42 chance of picking the second driver correctly.  When you calculate the probability of doing two things, you multiply the probabilities.  It makes sense that there ought to be less probability of picking two numbers in a row than of picking one, right?  So the odds of picking two drivers in the right order is 1 in (43 x 42) or 1 in 1,806.

Continuing this pattern…

# picked in right order

Calculation Chances are …
1 1 in 43 1 in 43
2 1 in (43 x 42) 1 in 1806
3 1 in (43 x 42 x 41) 1 in 74,046
4 1 in (43 x 42 x 41 x 40) 1 in 2,961,840
5 1 in (43 x 42 x 41 x 40 x 39) 1 in 115,511,760
6 1 in (43 x 42 x 41 x 40 x 39 x 38) 1 in 4,389,446,880
7 1 in (43 x 42 x 41 x 40 x 39 x 38 x 37) 1 in 162,409,534,560
8 1 in (43 x 42 x 41 x 40 x 39 x 38 x 37 x 36) 1 in 5,846,743,244,160
9 1 in (43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35) 1 in 204,636,013,545,600
10 1 in (43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34) 1 in 6,957,624,460,550,400

That’s one in 6.9 quadrillion to get all ten in the right order.

Is Picking Them in Order Harder?

What if TNT had just said you had to get all ten, in no particular order?

If you look at ten numbers, there are ten ways of picking the first number, nine of picking the second, etc. That multiplies out to there being (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1=) 3,628,000 different ways of organizing ten numbers in every which way possible.

If TNT had decided that you only needed to get the drivers right, but not the order, your chances would increase to a whopping 1 in 1,917,334,783.

But there aren’t Really 43 Drivers Capable of Placing in the Top Ten…

OK, in reality, the odds are a little better.  The calculation above assumes that the finish is a totally random event and we know that it’s not because there are 7-9 start and parkers.  Realistically, you’re picking from maybe 35 cars (8 S&Ps), so the odds for getting all ten in the right order if you’re only picking from 35 drivers are 1 in 818,441,006,423,040. or 1 in about 818 trillion.

But there aren’t Even Really 35 Drivers Capable of Placing in the Top Ten…

Yeah, the husband tried to make the argument that you’re really only choosing from about 17 or maybe 20 drivers.  Five words:  Regan Smith and Trevor Bayne.

Just for comparison…

Odds of being struck by lightning are 1 in 576,000.
Odds of being killed by lightning are about 1 in 2,320,000
Odds of a meteor landing on your house: 1 in 182,138,880,000,000

So you’ve got a better chance of a meteor landing on your house than winning that million dollars.

Often for promotions like this (free televisions if it snows 10 inches on New Year’s Day!!), a company will take out an insurance policy.  They’ll pay some amount of money to hedge against paying more.  The people at the insurance company who figure out how much to charge them use these types of calculations to figure out the risk.  I’m guessing TNT wouldn’t want to pay much of a premium because the odds are clearly in their favor.  But it’s a great promotion.

Does this mean you shouldn’t play?  Heck no – TNT isn’t charging you to enter, so get your best guess together and see if you can beat the odds.

RANDOM NOTES

Look at this cool project from Clemson and DuPont to take middle and high school teachers to the racetrack and teach them about science!  Way to go, Tigers.

The probability of becoming a saint is estimated at about one in 20 million, but if you’re Jacques Villeneuve, the odds rise to one in a flying pig.

Gratuitous link to The NASCAR Insiders just because their Wednesday Q&A is always worth checking out – it is a blog I always learn something from!

Daytona this weekend – read all about drafting vs. bump drafting, why you’re likely to see two but not three cars drafting together, why NASCAR limits radiator pressure to try to keep the two-car draft to a minimum, and why drivers shift to the right to get air to the engine if they’re turning left.  Or take a look at the Science of Speed video on drag and drafting.

 

100 Million vs. 78 Million is not the Numbers Question for FOX vs SPEED

The NASCAR pundits have again simplified a complex situation.  Incorrectly.

(Of course, at least they got the network right!  I got FOX and ESPN confused.  This is the problem with a 60-hour a week job and trying to blog about something utterly unrelated in the meantime.  My excuse is that I have a $3.5 million proposal due this week.  The same math holds, regardless of whether it is FOX or ESPN. Thank you Michael!)

The NASCAR Net is a-twitter since FOX floated a trial balloon about moving races from ESPN FOX to SPEED.  I’ve heard the argument over and over, in print and on radio that this is a bad idea because EPSN FOX is in 100 million homes and SPEED is in “only” 78 million homes.  They argue this would be a decline of 22 million potential viewers.  The question not being asked how many of those 22 million ESPN FOX watchers are actually potential viewers?

Point number 1:  Diehard NASCAR fans are going to find the race on television wherever it is.  Rabid fans are going to get whatever cable package they need in order to watch races, or they’re going to find a local sportsbar that carries the race.  Casual and incidental viewers are the ones that will make a difference in numbers.

Point number 2:  A very small fraction of households receiving a network watch it.  The highest rated race of 2010 on ESPN was August Pocono, with 6.3 million viewers.  Let’s assume an average of 2 people per household, so if ESPN is in 100 million households, that corresponds to roughly 200 million viewers.  ESPN pulled in 3.2% of the viewers who had the option of watching the August race at Pocono.

SPEED is in 78 million households, so assuming the same two people on average per household, there are 156 million potential viewers.  If SPEED captured the same 3.2% of their possible viewers, that would be 5.0 million viewers.  The difference is 1.3 million viewers — if you are willing to ignore point 3.

The numbers for FOX – let’s leave out the Daytona 500, which was 13.3 million and I bet FOX isn’t going to move that – are similar.  The highest rated race was April Talladega, with 8.45 million viewers.  Out of the 200 million possible eyeballs, that’s 4.2%.  4.2% of SPEED’s viewing audience is 6.55 million viewers, so again, we need an increase of about 1% to match FOX’s numbers.

Point 3:  Consider the demographics of FOX viewers vs. SPEED viewers.  SPEED is a motorsports channel.  I would think you’d be more likely to get a motocross fan to watch NASCAR than an average television viewer.  Which network is more likely to promote the race during other shows?  Which network is more likely to have the schedule freedom to do extended pre- and post-race shows?  All SPEED would have to do to equal the viewership from ESPN would be to attract 0.86% of the remaining viewers and about 1% to equal the viewership from FOX.  We’re really talking more like a difference of 2 million than 22 million.

There are many factors besides numbers, but numbers aren’t as big a factor as some are trying to make them out to be.

Just for fun, here are some stats for ESPN and SPEED viewership. They are from 2006-2007, but that’s the latest I have easy access to.

Category ESPN SPEED
Men 69% 80%
Women 31% 20%
18-34 28%
35-54 39%
55+ 33
18-49 69%
25-54 63%
$75,000/year + 43% 38%
$50,000/year + 62% 61%

When Oil Pans Put On a Little Weight…

According to @tomjensen100, the oil pans confiscated from the Joe Gibbs Racing cars weighed between 20-30 lbs.  Lee Spencer posted some nice pictures of them.  A normal oil pan weighs about 4-5 lbs.  Why in the world would you make your car heavier?  It’s not always the weight of the car, it’s how the weight is distributed in the car.

When a car accelerates (and by accelerates, we mean here the physics use of the word, which includes  braking, speeding up and/or turning), the way the weight of the car pushes on the wheels changes.  It’s called load transfer.  (P.S.  There is a difference between ‘load transfer’ and ‘weight transfer’.  Don’t get them confused around Bob Osborne.)

The body of the car is suspended (relative to the wheels) on springs and shocks.  When you brake, the body rotates a bit toward the front.  When you accelerate, it shifts toward the rear.  When you turn left, weight shifts to the right and when you turn right… well, we’re at Michigan this week, so we don’t really care what happens when we turn right, do we?

Shift happens because of torque.  Torque is the ability to rotate.  It’s like force, but slightly more complicated:  torque depends on force, but also on how far the force is from the rotation point.

Go to the nearest door and try to open it by pushing on it close to its hinges.  Doesn’t work so well, eh?  Push a little further away – should be easier.  (If it’s not, make sure you are using a door that does actually open outward.)  There’s a reason the handles of doors are as far away from the hinges as possible.

The torque you can exert on something is proportional to the perpendicular distance you are from the point about which the thing is turning.  When you use a cheater bar on a wrench, you’re moving the force (that’s you pushing or pulling) further from the nut you’re trying to turn, which makes it easier for you to turn.

Now pretend you’re trying to tip over a car.  (Insert your own snarky hockey comment here.)  You  have two choices – the car on the left and the car on the right below.  Their centers of gravity are marked by the circles.  Both cars weigh the same and you’re going to push from the left as shown by the arrow.  Which one is easier to tip over?

Imagine you're the arrow.  Which car is mostly likely to tip over?Hopefully, you pick the one on the right.  It’s center of gravity is further from the ground (the car rotates about the point where the right tire touches the ground), so it’s easier to tip.  It’s the same reason football players crouch down — it makes them harder to tackle.  The lower your center of gravity, the more stable you are.

When a car turns, it’s the same thing as applying force through the center of gravity.  A large force tips the car over, but a smaller force just rotates the body relative to the wheels.  If  car is turning left, the rotation of the car puts more force on the outside tires than on the inside tires.

Why does that matter?  The grip each tire provides is proportional to how hard it’s being pushed into the track.  The first law of racing is that you can only go as fast as your least grippy tire.  Maybe you have 1200 lbs of force on your right rear coming out of turn 4- if you only have 200 lbs pushing the left front down, you aren’t going to be turning very quickly.

You’d like to have all four tires pushed into the track with comparable amounts of force.  The less load transfer, the better.  The biggest complaint from the drivers when they switched to the new car was that the higher center of gravity made them feel like they had moved from sports cars to SUVs.

The center of gravity (CG for short) is determined by how weight is distributed in the car.  Everyone has to make the same minimum weight, but it makes a significant difference where the weight is located.  The goal is to get the CG as low as possible.  Teams have been making things like dashboards out of carbon fiber to decrease the weight and give them the opportunity to add ballast lower in the car.  A heavier oil pan would lower the CG, which would decrease load transfer and make the car faster.

Another possibility mentioned by John Darby (thanks to Bob Pockrass for noting it) is that the bottom of the oil pan could have been contoured to change the aerodynamics underneath the car.  You don’t want air getting under the car.  What air does get under there you want to flow smoothly out.  NASCAR doesn’t let the teams do much with the underside of the car – but it’s one of the few places where there is still some room to explore.

I remember asking a crew chief whether something was legal.  He squirmed a bit and finally said, “Well, it’s not illegal”.  NASCAR leaves grey areas – fewer and fewer of them, but they are there.  The crew chiefs and car chiefs scan the rule book trying to think of things they can do that are not specifically prohibited.  On the one hand, you can argue that running a piece that wasn’t previously submitted to NASCAR for approval is illegal.  But the rulebook (thanks Jeff Gluck) contains very little in the way of specifics for the oil pan.  It does, however, contain the magic phrase “must be acceptable to NASCAR officials”.

Why Six Points is About Right for What Used to be a 25-Point Penalty

I was watching the movie A Clockwork Orange the other night.  There is a scene where Malcolm McDowell, having been “rehabilitated” and returned to society incapable of defending himself, is being beat up by an old man.  He can’t even defend himself.  For some reason, it made me think of Kyle Busch.

To top off an already tough couple of weeks, Kyle’s car failed tech inspection (the front of the car was too low) after his third-place finish at Pocono last week.  NASCAR made a rare exception to their Tuesday penalty announcements due to the NASCAR Hall of Fame announcement being scheduled for Tuesday.  Monday, NASCAR docked the team 6 points and fined crew chief Dave Rogers $25,000.  Under the old scoring system, this would have been a 25-point penalty.

Graph of Old Points System vs. New Points SystemSix points seemed like an odd number (I know, it’s an even number – I mean odd strange).  Just out of curiosity, I graphed the new scoring system against the old scoring system (shown at right). I ignored bonus points because they are variable from race to race.  The bonus points put a little wiggle in the graph here and there (again, depending on which race), but they don’t change the overall conclusions.

The points toward the left and toward the bottom represent the worst finishes.  The last point in the upper right-hand corner represents the winner.  For most of the graph, the relationship between the old points system and the new points system is linear.  Using the handy formula y = mx+b, we can calculate that for the lower part of the graph, the slope (m) is 3 and the intercept (b) is 31.  Look to the bottom of the blog for a large graph showing the slope.

The 31 is simply an offset.  Under the old system, the lowest score you could get (assuming you were in the race) was 34.  In the new points, the lowest score you can get is 1 (1*3=3; 34-3 = 31 QED).  What we’re really interested in here is points relative to other people’s points, not points overall.  We could have taken the old system and subtracted 31 from everyone’s score and the results would have been just the same in terms of where people finished and how far away they were from the next person.  The relevant parameter here is the slope.

Notice that when you reach position 10 or so, the data start to deviate from a straight line.  NASCAR used to make progressively larger differences in points as you finished higher.  There was much more difference from 1st to 2nd than there was from 9th to 10th. That was done to try to reward drivers for finishing races.  The new scale is 100% linear and the motivation to win rather than place a comfortable second is that the last two spots in the Chase are determined by number of wins.

If you approximate a straight line going positions 10 though 2, you end up with a slope around 4-2/3.  So at the low end of the finishing order, one point in the new series is about three points in the old one.  If you look at higher finishing places, one point in the new series is about 4-2/3 points.  I drew in the slopes and made the picture bigger below to make it more evident.  You’ll notice that my straight line for the higher finishers is not a great fit due to the non-linearity.  I didn’t even try to include the winner.

Twenty-five points would correspond to anywhere from 8 points (using slope 3) to 5.4 points (or so, using slope 4-2/3).  So 6 points is on the lower end of the range, but it seems perfectly reasonable.  Too bad for Dave Rogers they didn’t scale the fine as well – he’d be ($25,000-$6,000=) $19,000 richer.

Plot of Old vs. New Scoring System with Slopes shown

 

 

Pocono: Slightly Shifty

The big news for Pocono is that drivers can shift…again.  Which brings up the obvious dual questions of: Why would you want to? and Why didn’t you before?

Compare how fast the wheels have to rotate with how fast the engine rotates.  Both are measured in revolutions (or rotations) per minute – rpms.  Assuming a tire circumference of 88.6 in, tires have to rotate from 417 rpm (at 35 mph), to 1490 rpm (125 mph) to 2146 rpm at 180 mph.  The graphic tachometer on television tells us that the engine runs between 7000 rpm and 9500 rpm most of the time.

Gearing for a Borg Warner MM6 manual transmission and a GU6 3.42 rear-end gear, as might be found in a Corvette.

You can’t connect the engine directly to the wheels because of the difference in rotation rates.  This is where the gears come in.  A car has two sets of gears:  The first I’ll talk about is the rear end gear, which I seem to remember is somewhere around 3.8 or 3.9 for Pocono.  The rear gear reduces the rotation rate coming from the driveshaft and sends that rotation to the wheels (as shown in the diagram).  A 4.0 gear would produce a rotation rate coming out of the gear that is 1/4th the rotation rate coming into the rear gear.  If the driveshaft is rotating at 5000 revolutions per minute (rpm), the wheels would be rotating at 1250 rpm. (A 4.0 gear would mean that for every four rotations coming in, one rotation goes out.)

With a 4.0 rear gear, your engine would have to change speed from 1600 rpm to about 8500 rpm going from 35 mph to 180 mph.  The problem is that an engine produces its maximum power over a narrow range of rpms.  (It also produces its maximum torque over a small range of rpms, although not the exact same range as the maximum power band.)  You’d like to have the engine operating in the target range all the time.

This is why you need a second set of gears, which are found in the transmission.  This series of gears (usually 4, 5, or 6 different gears) gives you different sizes so you can keep the engine running near its sweet spot — regardless of how fast you’re going.  Fourth gear on most transmissions is 1:1, meaning that there is no speed change through the transmission.  On a passenger car, like the one from the gearing figure, the higher gears (overdrive) reverse the ratio.  0.50:1 means that the rotational rate coming out is higher than the rotational rate going in.  NASCAR prohibits overdrive.

In trying to go faster and faster, teams were moving their engine’s target range to higher and higher rpms – which means higher and higher costs.  In 2005, NASCAR instituted a gear rule to keep engine speeds (and thus cost) down.  NASCAR gives you a limited choice of rear-end gears and dictates the transmission gears as well.  Those choices keep the maximum engine rotation rate below about 10,000 or 10,500 rpm without having to implement a difficult-to-enforce engine rule.

NASCAR changed the gear rule for Pocono this year.  First gear can be anything you want.  Second gear can be 1.70:1 or greater, and – this is the big change – the third gear limit changed from 1.28:1 to 1.14:1 or greater.  Fourth gear stays at 1.00:1.   (“or greater” means that the first number may be larger, but not smaller.)  NASCAR still doesn’t allow overdrive.  Normally, the rule book prohibits gears between 1.00:1 and 1.28:1 except for road course events.

Pocono - certainly one of the more unique tracks on the NASCAR circuit

Why Pocono?  Most oval tracks have four turns, with the frontstretch and backstretch close to the same length.  Pocono has three turns and three straightaways:  a frontstretch of 3740 feet, a backstretch (Long Pond) of 3,055 ft and a short straight of only 1,780 feet. You can imagine that the rpm the car reaches is very different coming down the two long straights (i.e. coming into turns 1 and 3) compared to coming down the shorter straight (i.e. into 2).  What you’d like is for the engine to be turning at about the same rpm into each turn.

It seems like NASCAR’s change is too small to be meangingful – from 1.28:1 to 1.14:1 is only 0.14, right?  Actually, it’s a factor of two.  What makes a difference is how much above 1.00 the gear is.  The important thing about moving from 1.14:1 to 1.28:1 is moving from 14 to 28.

For the sake of argument, let’s say the engine is ideally in 7200 rpm in fourth gear.  When you shift to third, a 1.28:1 gear (which used to be the lowest for third), requires the engine to run at 9216 rpm (=1.28*7200) to maintain the same speed.  That takes you far away from the best rpm range for your engine.  Changing from 1.28:1 to 1:14:1 means that third gear only requires your engine to run at 8208 rpm.  That may seem like it is still a big shift; however, given the way the power and torque curve vary with rpm, it’s small enough to mean that you’re close enough to your power band for it to work. It’s a shift of about 1000 rpm instead of 2000 rpm with the 1.28:1 gear.  That gives the engine shop – and the driver – some interesting options.

This type of a rules change is, in my opinion, exactly the direction NASCAR ought to be moving to open up areas for people to be innovative.  It’s a relatively minor change in terms of enforcement.  It keeps the teams from pushing into the higher rpm ranges (and thus steeply pushing up engine costs), but it allows the engineers and the drivers to pursue different strategies.  For example, most drivers will be shifting in turns 1 and 3, but others (like Denny Hamlin) plan to shift only in turn 1.  Another aspect is how shifting affects fuel mileage.  Overdrive gears are there because the more rotations an engine makes, the more friction it has to overcome.  And, as Carl Edwards points out, every time you shift, you run the chance of screwing up and damaging the transmission.  Most NASCAR drivers aren’t used to shifting this much during a race.  Do you try for what might be a small advantage and shift at the cost of possibly screwing up the transmission?  Do drivers like Marcos Ambrose, who have a lot more experience shifting, have an advantage?  Does the engine shop adapt different strategies for drivers who are comfortable shifting compared to those who are not?

Unfortunately, this rule really makes a difference only at Pocono due to it’s unique configuration.