The main reason that matches consist of multiple games is to increase the likelihood that the best team will win the match since the worst team always has some probability of beating the best team in any given game.

Not surprisingly, the more games that constitute a match the greater the likelihood that the better team will win the match. But if the teams are almost evenly matched, it can take a large number of games before the likelihood that the better team will win the match becomes significant.

But how many games is that?

If two teams play a match (in which the team winning the majority of games wins the match and tied games are not possible) consisting of an odd number of games, N, in which the better team has a probability of winning p, how many games must be in the match before the better team has a probability q of winning the match?

Let's go for a couple of specifics.

Case #1: Team A has a 60% change of winning a given game. How many games must be in the match before they have at least a 90% change of winning the match.

Case #2: Team A has a 51% change of winning a given game. How many games must be in the match before they have at least a 99% change of winning the match.

 

Case #2: Team A has a 51% change of winning a given game. How many games must be in the match before they have at least a 99% change of winning the match.

I haven't worked any of the math but a similar example was used in a stats class. The professor showed that a 51/49 advantage at Wimbledon virtually ensures winning. (I take for granted he was talking about the round-robin, not the elimination tournaments.)

To win a round-robin tournament, the winner needs to win every match he plays. Everyone else loses exactly one match. A Wimbledon men's match is the best of 5 sets. To win a match you'd have to play and win at least 3 sets of at least 6 games per set, or 18 games total. With only a slight advantage, you'd end up playing many more than the minimum number of games, since your opponent would win a few. If you play all 5 sets to a near tie (such as 6-4), that's 50 games!

Related: https://plus.maths.org/content/puzzle-page-95

 

I haven't worked any of the math but a similar example was used in a stats class. The professor showed that a 51/49 advantage at Wimbledon virtually ensures winning. (I take for granted he was talking about the round-robin, not the elimination tournaments.)

To win a round-robin tournament, the winner needs to win every match he plays. Everyone else loses exactly one match. A Wimbledon men's match is the best of 5 sets. To win a match you'd have to play and win at least 3 sets of at least 6 games per set, or 18 games total. With only a slight advantage, you'd end up playing many more than the minimum number of games, since your opponent would win a few. If you play all 5 sets to a near tie (such as 6-4), that's 50 games!

Related: https://plus.maths.org/content/puzzle-page-95

I don't follow. My understand of a round robin tournament is that everyone plays every other player the same number of times. So if I have five players in a single round-robin tournament, the why couldn't the final win-loss stats be

A 3-1 (beats B, C, and D)
B 1-3 (beats E)
C 2-2 (beats B and D)
D 1-4 (beats E)
E 2-2 (beats A and C)

 

I knew I was opening a can of worms with my poor understanding of tournament structures, or Wimbledon. I just remembered the point that enough trials can detect even a small advantage.

I actually used this on the job years later to explain to my boss why doing a few lab tests was never going to reveal the small effect we were looking for. I did the full math on comparing samples from two bell curves that nearly overlap (such as the heights of men versus women).

It was completely lost on him.

 

I knew I was opening a can of worms with my poor understanding of tournament structures, or Wimbledon. I just remembered the point that enough trials can detect even a small advantage.

Definitely agree with the conclusion. I don't follow tennis so I don't know what their tournament structure is like, but it goes on long enough that I can believe it has, in theory, good discrimination. Of course, it's not so much the total number of games but rather the number of games per player that is the big factor.

I actually used this on the job years later to explain to my boss why doing a few lab tests was never going to reveal the small effect we were looking for. I did the full math on comparing samples from two bell curves that nearly overlap (such as the heights of men versus women).

It was completely lost on him.

I had a somewhat similar situation, but different outcome. We were designing a chip for Sony (or rather our customer was, we were designing the driver IC). The chip had 256 10-bit channels on it. The DAC need to be piecewise linear with sixteen adjustable breakpoints per channel and it was critical that the DAC be monotonic. Their approach was to put a huge amount of effort into a single DAC and then multiplex it to all the channels. They had failed to get yield on their parts because the tolerances were just too severe. So our customer convinced them to come to us and we used a separate and quite simple segmented DAC in each channel that was made up of 16 6-bit DACs, which were in turn made up of 6 1-bit DACs. The design fundamentally guaranteed monotonicity. But with 24,576 1-bit DACs on the chip, Sony was quite reasonably concerned about yield (they needed about 70%) since ALL of the DACs needed to work within spec, especially since they couldn't get a single DAC to yield anywhere near that.

So the obvious approach was to do Monte Carlo simulations in PSIPCE to get the expected yield. You can't do enough sims on the entire chip, a single channel, or even a single 6-bit DAC (at least not back then, which is about 15 years ago). So we had to do it on a 1-bit DAC. The problem was that I discovered that PSPICE was hard limited for three-sigma variation on the device parameters and I needed something like 5.2, if I recall correctly. So instead I set about to collect enough data from the simulations so that I could estimate the standard deviation of the performance of the 1-bit DAC and see if a DAC that was 5.2 sigma out would be within spec.

To make matters worse, we were using a digital process for a mixed signal design and so we didn't have hardly any matching parameter data. So we used the data from the analog process and degraded with a fudge factor based on out in-lab experience with other designs on both, which wasn't a whole lot.

This was a big enough concern for Sony that they flew four or five of us out to California (where our customer was located) and then we spent an entire data on a teleconference call with Sony. Of course, this was but one issue that was being covered, but I spent about two hours walking through the basis for my simulations and the justification for why they would, given accurate input data, yield valid results plus the data itself, my estimate of the uncertainty in the standard deviation that I got, and the final result -- and they asked some very pointed questions.

The results of the analysis were that we anticipated a yield of 78%, though I made darn sure to let Sony know that there were real holes in our base data that I simply couldn't put numbers to. But it was still enough to convince them to proceed. The chip yielded 82% on first silicon. The fact that we hit it that close owed a hell of a lot to luck, but the analysis put us in the ballpark well enough for luck to have a chance.

In addition to soothing an anxious customer, the work had real benefit to the design. We took into account as many different things as we could think of including device geometry and performance parameters, but also routing variations and capacitive coupling. With a very simple 1-bit DAC (very, very few transistors, though I don't recall how many) you can through a LOT of stuff at the circuit model. An analysis of the data that I dumped from the sims allowed me to quantify how much each variable contributed to the total standard deviation and we were surprised when some of the things that we thought would be minor turned out to be significant factors and some of the obvious variables contributed very little. So we spent a couple days tweaking the layout to reflect that information.

Another thing that struck my at that meeting. I was also teaching at the local university and had just sat through a bunch of senior design presentations and there was so much emphasis placed on having engaging, colorful, animated presentations and whoa be to the team that just had a bunch of black and white slides with some words and graphs on them. Yet here we were doing exactly that for a presentation to one of the 800-pound gorillas in the industry and they didn't bat an eye. Over the years I've come to the conclusion that, more often than not, whizzy presentations are a strong indicator of a lack of technical merit and vice versa.

 

... we were surprised when some of the things that we thought would be minor turned out to be significant factors and some of the obvious variables contributed very little.

if I had a penny for every project in which that has happened to me too...

... whizzy presentations are a strong indicator of a lack of technical merit and vice versa.

I know what you mean, but most customers expect a minimum (but relevant) number of graphics to help them visualize what exactly your point is... and that's the bottom line, they're evaluating *you*, not your presentation ... as you've just said, the key is not to overdo it.

 

if I had a penny for every project in which that has happened to me too...


I know what you mean, but most customers expect a minimum (but relevant) number of graphics to help them visualize what exactly your point is... and that's the bottom line, they're evaluating *you*, not your presentation ... as you've just said, the key is not to overdo it.

Oh, to be sure we included relevant graphics. Usually nothing more than screen captures of schematics, simulation results, or sections of layout. When we had an Excel plot or something like that it was almost always bare bones.

One very valuable lesson that the boss there insisted on was no "horse slides", in which you have a picture of a horse and the slide title says "A Horse". He wanted the slide titles to, as much as possible, get across the key central point of the slide. If that wasn't possible, then at least pick something interesting and relevant, but don't just tell the audience what is already glaringly obvious. It was a real pet peeve of his and it was something that most of us where glaringly guilty of -- at first.

For example, if we had a graphic of the unit cell layout, we did NOT title the slide "Unit Cell Layout". Instead, it might be something like, "Unit cell layout allows 2-D mirroring to share resources and increase fill factor."

At first we all pretty much hated putting up with the boss about this, but it didn't take long before he won us over. Meaningful and impactful slide titles really do add to the presentation. But perhaps even more valuable is they force you to give some serious thought to why you have a slide in the first place and what it is that you are trying to get across to the audience and, with that in mind, let's you focus on making sure your presentation delivers on that point.

It also puts the punchline right up front (which, of course, means that it isn't a punchline anymore). Without it, people have a tendency to want to "set up" the audience to prepare them to be wowed by the result. But in a technical presentation, that shouldn't be the goal. The goal should be to establish your claim up front and then proceed to justify it to the customer. By seeing the assertion right away, the customer is listening to your presentation from the perspective of deciding whether they buy your claim or not as you are putting forth your case, not in retrospect.

 

To get back to your questions:

Case #1: Team A has a 60% change of winning a given game. How many games must be in the match before they have at least a 90% change of winning the match.

Case #2: Team A has a 51% change of winning a given game. How many games must be in the match before they have at least a 99% change of winning the match.

Can it be as easy as using the formula I wrote in the 2/3 example

MWP= (2LP x HP^2) + HP^2 where MWP=the match win probability, HP= is the high probability of winning a game, or edge, and LP, or 1-HP, is the low probability of winning a game.

Except we substitute (n) for 2 and, by using the match win probabilities that you provide in the question, solve for n...(and then add 1).

So,
.90=(n.40 x .60^2) + .60^2

?

I guess I could test this out myself, but I see on my schedule that I now have "Executive Time" for the next few hours - sorry

PS: I don't remember any of those rules for solving linear equations and I don't find relearning them as desirable as going to get some food.

PPS: I saw your solution to the 2/3 - of course it is correct. I also saw the formula and thought ohhhhhh that's the way it should be written. But I feel no embarrassment - thinking Dr McCoy saying dammit Bahn, I'm a Neuroscientist, not a Mathematician!

Seriously, this is fun stuff and I would love to see a regular feature of original problems along the lines of the old "Mathematical Recreations" in Scientific American that I enjoyed so much whether I understood them fully or not.

 

So,
.90=(n.40 x .60^2) + .60^2

So, please solve for n to the nearest thousandth of a game.

 

So, please solve for n to the nearest thousandth of a game.

Sorry, but no thanks.

 

Wolfram Alpha is your friend:

 

Wolfram Alpha is your friend:

Looks like a friend that I might like.

 

Hey, this is a cool program for mathematical retardates like myself. It, literally, means I can generate errors with a frequency of at least two orders of magnitude greater than baseline!

To go back to my post:
"
Can it be as easy as using the formula I wrote in the 2/3 example

MWP= (2LP x HP^2) + HP^2 where MWP=the match win probability, HP= is the high probability of winning a game, or edge, and LP, or 1-HP, is the low probability of winning a game.

Except we substitute (n) for 2 and, by using the match win probabilities that you provide in the question, solve for n...(and then add 1).

So,
.90=(n.40 x .60^2) + .60^2"

Which is 3.75 + 1...Five games.

Case #1: Team A has a 60% change of winning a given game. How many games must be in the match before they have at least a 90% change of winning the match.

5 (wheres the click for the WRONG buzzer sound?)