The Price is Right sure is an old show. Unfortunately, whenever I see the show it brings me back to my childhood. Not the good childhood, instead the days of me being home from school sick. The show came on during the day, so the only time I would see it was if I was home sick. I guess I am like Pavlov's dogs. I guess one reason for this response is that the show's set and gameplay have not changed since the '70s.
And this bring us to the Price is Right wheel. The basic idea is that contestants spin it and try to get a sum close to $1. Why am I thinking about the wheel? I blame Dan Meyer. Dan made this very nice video showing how the wheel beeps as a function of time.
So, here is the question: Can I come up with a strategy to make the wheel land at a particular spot? Clearly, there are a couple of things: Where does the wheel start? Where do you want it to end? How fast do you have to spin it and where do you let go?
Collecting Data
If you don't know Dan Meyer, you should. He is a master at making math content interesting and pretty. Although I think his video has a very visual appeal, it isn't too useful. The first thing to look at is the angular acceleration of the wheel. I suspect this is a constant value, but I need to find out first. In order to look at the angular acceleration, I need the angular position of the wheel as a function of time.
If you want to just get the angular position of the wheel using video analysis, you would have some problems. You can't always see the wheel from the best angle; I really don't think this would work too well. Dan took the approach of just listening of the "beeps." A different solution would be to use video analysis and just mark the frames where the beeps would be made. the wheel has 20 sections. This means that the angular distance between each "beep" would be 2 π divided 20 radians (0.314 radians).
So, repeating Dan's experiment, I get the slightly more useful graph (oh, and this is AFTER the guy let go of the wheel).
My method seems to work. Great.
Angular Acceleration
You could try to make this problem super complicated (which is usually the path I choose). However, in this case it might be easier to just see if the the wheel's angular speed decreases at a constant rate. Here is a quick angular kinematics refresher.
If I call θ the angular position of the wheel, then I can define the average angular speed (ω) as:
But what if the angular speed isn't constant? What if it is slowing down like the Price is Right wheel? Well, in that case I can also look at the angular acceleration (α):
Here, ω1 is the angular velocity at the beginning of the time interval Δt and ω2 is the angular velocity at the end. Perhaps you can see where this is going. This all looks just like the equations for constant acceleration in one dimension (kinematics).
OK, angular acceleration is fine and all – but the problem is that I don't know this value. I have values for angular position and time. Let me re-write the angular position (θ) in terms of the average angular velocity.
If the angular speed changes at a constant rate (constant angular acceleration), then I can write the average angular velocity as:
This means I can re-write my expression for the final angular position as:
Now, I can use the angular acceleration to remove the final angular velocity:
BOOM. There is your famous kinematic equation. No calculus required.
Angular Acceleration of the Wheel
The above derivation assumes constant angular acceleration. But does the wheel have a constant acceleration? If it did, a plot of position versus time should be a second order polynomial. Well, I can fit a second order polynomial fit to the data that I have. For this, I can use the polyfit function in python. This is what I get:
Quick reminder: When you fit a polynomial to data (in python or Excel), you will get three coefficients. One of those coefficients will be a number that goes with the t2 term and it will often be labeled as "a." This is not the acceleration. This matches up with the (1/2)α term – so the angular acceleration will be twice this coefficient.
But what can I say about the angular accelerations for this data? First, the blue dots are the actual data points and the red line is the plot of the fitting polynomial – the eyeball test seems to indicate a pretty good fit. Another positive indication is that the angular accelerations for the two spins are similar (-0.143 rad/s2 and -0.145 rad/s2). So, is it constant? I NEED MORE DATA.
I couldn't help myself. I went and looked at several more Price is Right wheel spins. Why do people put this stuff on YouTube I'll never know – but thank you. Here is histogram of the angular accelerations of different spins.
This gives me an average of -0.1701 rad/s2 with a standard deviation of 0.0216 rad/s2. Honestly, I think the video quality on some of these videos is a little questionable. It is possible that the frame rate is off or some frames are skipped. I had one video with four good spins and they were all right around the -0.16 rad/sec2 range. It is also possible that the friction in the wheel changes with weather conditions. Maybe they lubricate it from time to time.
So, is the angular acceleration constant? For a particular spin, this seems to be the case. For spins on the same day this also is likely true. Now, just for comparing distributions, what do the starting angular speeds look like? Here is a histogram for that:
The starting angular speeds are a little more spread out (as you would expect since they are from different people). Just let me say (for no real reason) that the average was 1.82 radians/second with a standard deviation of 0.346 radians/second. Also, I suspect these speeds are bunched a little artificially due to the frame rate of the videos and the discrete nature of the angular measurements (in segments of 0.314 radians).
For the rest of the post, let me assume that the angular acceleration of the slowing down wheel is indeed constant.
Predicting the Result
This is probably what you want, right? If the angular acceleration (which I will now refer to as just the acceleration because it is short) is constant, can I predict the final angular position of the wheel? Obviously, it depends on how fast the wheel starts to spin. If I look at this equation, I could get the final angular position if I knew the time.
Alas, that is not something I would know before I spin the wheel. But what I do know is the final angular velocity and the acceleration. Since the wheel stops, I can write:
Now, I can substitute this in to the first angular equation to eliminate the time dependency, I get:
So, there you have it. If you need to know how fast to spin it, you just need the angular acceleration and the angular distance. In theory, this can be accomplished.
Practical Considerations
Now for the tough part – reality. I see two big problems. Problem one: Where do you start? Oh sure, you just spun and it landed on 40 cents or something. But where is the wheel when you let go? From the Dan Meyer video, it looks like the first guy releases the wheel when it is four segments (1.26 radians) past the starting point. You can't see his second spin. Looking at other videos, it looks like people release the wheel anywhere from two segments (0.628 radians) to 4.5 segments (1.43 radians). Everyone has their own preference, but how precise could you be?
The second problem is angular speed. Suppose you calculate an initial angular speed of three radians per second. How precisely could you launch the wheel with this speed?
Let me go ahead and simulate this spinning. I will just guess that you could launch the wheel with an accuracy of +/- 0.1 radians/sec. (Yep, I just totally made that number up.) For the release angle, this might be easier to control. Let me guess that you could release this within 0.05 radians of where you want. (You could use the distance to the floor as a reference.)
If I spun the wheel 100 times, I would expect a normal distributions of speeds with a standard deviation of 0.1 radians per second. This is essentially what is done with the Monte Carlo method for uncertainty. Here is a plot showing the normally random distributions of these spins. Oh, let me just say that I want to start with an initial angular speed of two radians per second.
Is it OK to model this as a normal distribution? Who knows. Really, the only way you could be certain is to look at a whole bunch of spins (way more than 10 or so). In general, the normal distribution tends to give pretty good results for stuff like this.
Now that I have a normal distribution of both starting positions and starting velocities, I can calculate the ending angular position for these 100 spins. This is what the distributions of final angular positions would look like:
Out of these 100 spins, the average spin angle would go 11.76 radians with a standard deviation of 1.3 radians. But what does this all mean? Is this good enough to get where you want? Well, here is the trick. Remember that the angular size of one "segment" is only 0.314 radians. This means that of these 100 spins, maybe about 20 would be within the target range (of course probability is involved here, so this number can vary). Let me look at 5,000 spins. In this case, I have 533 spins that landed within +/- half a number segment on the wheel.
What Does This All Mean?
I think, in terms of the game, you would just have to spin the wheel and hope for the best. The problem is that variations in the initial conditions are great enough that you coudl easily be off by 1 wheel segment. This is a big deal since the numbers in adjacent segments are quite different.
