Tuesday, January 27, 2015

Pro Sport Team Mobility

Oh, the Oilers...

As of today, with a little less than half of the 2014-2015 NHL season remaining, the estimated odds of the Oilers getting into the playoffs are approximately 0.008%. They're currently last in their division, and the estimated chance of them staying there is 70.7%. Congratulations on another wonderful season!

The only thing the Oilers have going for them is that the NHL draft system gives a bonus to teams that do badly, with the hope of eventually balancing things out. Of course, the Oilers have supposedly been on the receiving end of this for a few years without success, but maybe this time it'll actually work?

But exactly how well does does the draft system work for helping the worst teams out in future seasons? I decided to find out.

The NHL has only had 30 teams playing since the 2000-2001 season, so I decided to stick with the years since then. I started by looking at the teams that ended up in either the top or bottom quintile (20% of teams), and tracking how likely they were to make it into either category over the following five years:

For example, the above graph shows that a team that finishes in the top 20% of the NHL one year has a ~50% chance to make it back in the top 20% within one year, and an 80% chance to make it within 5 years. On the other hand, the worst teams in the league only have a 7% chance to make it to the top within one year, and only a 50% chance to make it within 5 years. Cool, right?

Reversing the situation looks like this:

This results in somewhat of a different trend. Teams that do poorly have a 2/3 chance of being in the bottom fifth of the NHL again within 3 years, but after that it plateaus and there doesn't seem to be much increased risk of them doing terribly. Also, it's about half as likely for a great team to end up doing terribly at any point within the subsequent 5 seasons as it is for a poor team to do awesomely.

These results lead into a discussion of how closely correlated a team's performance is year-to-year. Plotting how successful they are one year against the subsequent year looks like this:

Definitely a correlation, but nothing worth placing a bet on. This says that, in general, the teams that did well one year are likely to do well the next. No huge surprise there. Looking further down the road:

Five years later, and there's almost no correlation at all. This is the sort of thing that ought to make Oilers fans happy, if only it weren't for the fact that they've been in that bottom 20% for four of the last five seasons. Ugh.

The change in correlation between seasons is shown pretty well by this:

Which is an excessively pretty and smooth trend between seasons. This apparent regression to the mean for teams on the whole, though, also applies really well to the teams on the extremes (the best and worst of any season):

Suggesting that, surprisingly, the best and worst of the league will be statistically equivalent after only 3-4 seasons.

So does this give us much of a prediction for when the Oilers will finally start showing up to play real hockey? Not really, though historically the trend has been that they ought to have a 50% chance of making it to the top of the league within 5 years, and that they'll be about the same as today's best in the league (eg Anaheim?) within about 3 years.

The final question then becomes just how much of this effect is caused by normal luck and regression to the mean, and how much is influenced by player trades and the draft system? Turns out that it's likely quite a bit - comparing four major pro sports leagues on of the charts above gives the following:

(Chart was made using 13 NHL seasons, 12 NFL seasons, 10 NBA, and 15 MLB for consistent league sizes)

The fact that there's such a variation between sports suggests that there's more to inter-season variation than just random chance, which is certainly promising news for all those coaches and general managers out there. It isn't terribly surprising that the correlation values for NFL are consistently low, since the number of games each team plays is drastically lower than the other three pro sports leagues. On the other hand, the results for the NBA are rather surprising - teams tend to do almost the exact same one year as they do on the next, but tend to have an inverse correlation five years down the road. If the Oilers had been an NBA team, they wouldn't expect to stay at the bottom for very long.

So while this year looks like another dud for the Oilers, there's always hope. Plenty of teams have broken their slumps before, and it's hopefully only a matter of time before the Oilers have their chance.

Wednesday, December 17, 2014

My Issues with the Broadbent Institute's Inequality Report


Apparently I have it in for think tanks or something. Every few months a think tank somewhere comes out with a report that means well, portrays a message with fundamentals I agree with, but manages to mess up some amount of the data handling in a way that gets me riled up.

This time it's the Broadbent Institute. They released a report on income inequality recently, and presented the data in a virtually identical format to an American video from two years ago. While I agree that income inequality is a big issue in Canada, and I'm sure that the average Canadian isn't clear with just how bad it is, I have a pretty big issue with the statistical rigor in their report.

This is a screenshot from their video:

Along the x-axis, they have different population percentiles in 10% chunks. The chunk on the far right represents the richest 10% of the population, the one to its left is the next 10% richest people, etc.

The problem with this chart is that it shows the 50-60th percentiles as being richer than the 60-70th and 70-80th. They're trying to tell us that the 5th richest group is richer than the fourth and third richest groups. 

What!? That doesn't make any sense by definition.

These values appear to come from this table in their report:

Somehow, Canadians apparently consistently think that the middle 20% of the population is supposed to have more money than the 2nd wealthiest 20%. That's not possible, and I can't believe that it got all the way into the report and the video without anyone hitting the emergency stop button. The income curve shown in the first figure ought to be a version of a Lorenz curve, and necessarily should increase from left to right. Even if that is the actual result from the survey, it shows that either the survey wasn't clear enough in its instructions, or that adequate controls weren't in place in the survey to ensure accurate results. 

When I brought this up to them on twitter, their response was:

Which is... silly. There's a logical distinction between a "strong middle class" and a "middle class that's stronger than the upper middle class." They've clearly decided to ignore this distinction.

Finally, a (more than average) nitpicky point. Take one more look at this graph (where the blue line was the original "ideal" values);

The blue line of ideal values has five data points in it, which is great, but they're nowhere near where they should be! Each point (or kink in the blue line) corresponds with a 20% chunk of data, so they might have shown the mid-point 10%, 30%, 50%, 70%, and 90% marks. But instead they've shown the 0%, 25%, 50%, 75%, and 100% marks. This implies that the 0th percentile Canadian (the absolute poorest person in Canada) ought to have the wealth value that the people surveyed thought belonged to the bottom 20% as a whole.

Anyway, the point of all this is that research into wealth inequality is really important, and doesn't deserve to be handled quite this badly. If you're going to be sharing this report, please do so with a grain of salt.

Monday, December 15, 2014

Winter Tires in Canada

Well there's snow on the ground and the temperature's pretty low, so we can pretty solidly declare that winter is upon us. And with wintry blizzards comes one of the great Canadian traditions: changing over your summer tires for winter tires.

If you're anything like me, you probably waited until just after the first major snowfall to remember to put them on. This often ends up with you driving around dangerously for a week waiting for your appointment, all the while dodging other summer-tire skidders. It's a fairly dangerous and unpredictable way to go about driving.

Recently I tried looking up recommendations for when to put on your tires and came to an interesting discovery: almost every single source recommends to put them on once the temperatures dip below 7 degrees Celsius. Everyone from the tire producers to the Tire and Rubber Association of Canada agrees with this fairly precise temperature recommendation.

Why? Turns out that summer tires are made of a different rubber that gets quite stiff below 7 degrees, and reduces the friction of the tires (the comparison that was used was that they approach the consistency of a hockey puck). Winter tires become more effective below 7C, even on dry clean pavement.

Not to scale. Probably.
If you're looking to drive as safely as possible (which you should, seeing as road injuries are the 9th leading cause of death worldwide), it might not be quite enough to just wait until the forecast predicts a temperature below 7, seeing as it often takes time to book an appointment and by that point it could be a bit late. Fortunately, Environment Canada has the daily temperature for various cities over the past several decades all neatly stored online.

So I decided to take a look. These are the average mean daily temperatures for Edmonton per day for the 30-year span between 1981-2010:

Since each day of the year has a decent variation to them, it's also possible to determine the expected probability that any given day will be below 7 degrees Celsius (using their averages and standard deviations). That might get you something like this:

Once you have this, it's fairly straightforward to choose when to put on your winter tires. If you were willing to accept a 50% risk of being ill-equipped for the weather, you'd be looking to put them on sometime around the beginning of October, and take them off around the beginning of May. That's vastly longer than I typically have mine on for, and I suspect that's the same for many people. In total, an Edmontonian ought to have their winter tires on by October 1st, and leave them on for 210 days (at least seven months of the year!).

Of course, a 50% risk of having the wrong tires might seem a bit high for some people. If you were only willing to accept a 10% risk, you'd be looking at 261 days of winter tires starting September 4.

So that's all well and good for Edmonton, but how about the rest of the country? I decided to look at 30 stations' worth of data spanning 1981-2010 (~330,000 data points) to try to develop a map for winter tires in Canada. These stations included all major cities and a few select points to accurately represent the geographical differences. This is what I got:

Unsurprisingly, the northern territories tended to need winter tires more than the southern provinces (quite frankly, it's not worth taking winter tires off if you live in Iqaluit). What might be surprising to some is that even the warmest parts of the country, that hardly ever see snow, ought to have proper winter tires on for at least a third of the year.

Another way to represent the data is to show the probability of being below 7C on any given day like this:

Where green means 0% chance of being below 7C, and red means 100%.

The vast majority of Canadian cities have a high risk of being below 7C sometime in October, and it's important to know when exactly that will be in order to be sure you're driving with the best equipment available. In fact, the above graph can be summarized as follows:

One final thing to note: only the province of Quebec has legal requirements for winter tires, with the exception of some British Columbia highways. These legal requirements fall way outside of the 7 degree recommendation though. It's all well and good to have laws for additional safety when operating motor vehicles, but if they fail to capture the designed temperature ranges of the actual tires, it seems like a bit of a missed opportunity.

Monday, December 8, 2014

2014-2015 ski season

In case you haven't noticed, it's snowed a bit recently in town. And any time it snows in Alberta, I get excited that it's likely been snowing up in the mountains. And that means skiing!

As of December 7, the website OnTheSnow shows that Marmot Basin (the closest ski hill to Edmonton) has a snow depth of 90 cm. That sounds rather decent, and certainly right at the end of November it got a massive dump - but how does that actually compare to normal? I decided to figure out.

Here is the cumulative snowfall of Marmot basin for every ski season since 2007-08:

Alright, so there's quite a bit of variation in there. Maybe a better way of looking at it is like this:

For these graphs, the grey zone represents the maximum and minimum values over the last seven seasons, the light grey line is the average, and the black line is this season so far.

So there's good news and bad news here. The good news is that there's actually quite a bit more snow this year so far than normal! In fact, there's about as much snow at Marmot right now as there typically is by about January first. All in all, maybe not a bad time to go there, in fact!

The bad news is that, apart from two huge dumps, there really hasn't been any action in Marmot. It was way below any of the previous seasons measured until two weeks ago. Marmot looks like it's in a good position now, but if it hadn't gotten luck at the end of November it would pretty much just be rocks. In fact, we can tell it *has* been lucky - Marmot Basin typically only gets two to three snowfalls exceeding 20 cm per day per season (actually 2.43 average), and has already had two this year. Lucky for it now, but it's hard to predict for the future of the season.

Marmot Basin is also relatively easy to predict - on average by the end of the season, its total snowfall has a coefficient of variation of 37.1%. It also has a reasonably early season, with 100 cm of snow fallen on average by December 31.

But how about other Alberta ski hills? Take Sunshine Village, for instance:

Sunshine has a similar situation to Marmot Basin. It's been lagging behind previous years until the end of November (though still within normal ranges), and is now pretty much back on track. Hard to say how that will hold up though. They don't typically reach 100 cm of snowfall until a bit earlier than Marmot (average December 18), and tend to be more predictable (coefficient of variation of 23.8%). They also get far more snow in total than Marmot Basin does...

Lake Louise enjoys a base of 100 cm on average by December 16, but is raucously tricky to predict (coefficient of variation at end of season of 44.3%). Lake Louise has had the same problem as Marmot Basin - it had far less snow than previous years up until a sudden burst rather recently, but it's been flat since. Hopefully that isn't terrible news for the season...

Nakiska's almost doing the best for this time of year out of any of the last 7 years! Good for it. They tend to have more variation at this time of year than other Alberta hills too, so it's actually a bit tougher to say if they'll have a good season or not. They tend not to get a 100 cm base until around January 23rd, and have very unpredictable seasons, with a coefficient of variation of total snowfall of 48.2%.

Norquay's a bit sad. They're well within previous years' ranges, but it's still not looking nice. They'll get their first 100 cm on average by February 10 (yikes), and have a variation in total snowfall around 37.6%. Some years they don't even get 100 cm of snow, though.

Castle Mountain's another sort of sad mountain with a later season (100 cm average by January 11th) and high variability between seasons (43.2%). Both Castle and Norquay seem to have missed the awesome snow dump that the rest of Alberta had, but are tending to stick a bit better into where they'd be expected at this point in the season.

So overall for mountains in Alberta, it's looking like now is a great time to go to Marmot, Sunshine, Lake Louise, or Nakiska. They're certainly at least all doing much better than average for this time of year, and will likely continue to be above average for the rest of December.


Earliest decent season: Lake Louise (December 16)
Highest average snowfall: Sunshine Village (486 cm by May)
Most predictable: Sunshine Village (23.8% variation by season)

The sad thing is... BC mountains do way better on almost all counts. Take for example Fernie:

(100 cm by Dec 22, average snowfall 705 cm, COV 25.8%)

Or Whistler:

(100 cm by November 24, average snowfall 796 cm, COV 27.7%).

Both mountains consistently and reliably get far more snow than anything in Alberta. While that may make them sound great on paper, they still haven't had the trend-bucking dump that Alberta mountains have had, and are currently lagging quite far behind their Alberta peers. So while I can't guarantee that they'd have particularly good December skiing this year, you certainly ought to be able to rely on them for quality skiing in the mid- to late-season!

Monday, November 24, 2014

Foo Fighters Frenzy

This weekend, Edmontonians got the chance to stand in line for the Foo Fighters ticket pre-sale. The way the line-up system worked, though, left a few people upset - instead of a good old-fashioned first come first served system, Northlands used a lottery system where customers were given a number when they arrived, and then at a fixed time people were stopped from getting in line and a random number was chosen.

The person holding that number became the new front of the line, and everyone ahead of him was moved to the back in their original order. People were supposed to be given numbered tickets randomly, but instead they were given tickets in the same order that they arrived in.

It was billed as a process to avoid making people camp out early, as there was no benefit to being first in line, but sadly not everyone read the rules. This resulted in a system where some people had waited for hours to be at the front, only to find out they were likely going to be moved to the back of the line once tickets actually went on sale, and would have to wait even longer for less selection.

So this sounds like something that could be fun to analyze! Based on the information in the Journal article, and from a friend who was in line, I've come up with the following parameters for this problem:

  • 600 people join the line
  • They join the line at approximately a steady flow between 6:30 am and 9:30 am
  • My friend waited ~4 hours once the sale actually started, and had 490 people ahead of him at that point, so people spend an extra approximately 30 seconds in line for every person ahead of them after the lottery draw
  • There's some benefit to being the first person to pick a ticket, and people get less happy as less selection becomes available

Based on this, I came up with a utility function for every spot in line starting out. The utility function is 1.0 (perfectly happy) for someone who shows up right when the doors close, and by luck gets the first choice (so no waiting at all), and is 0.0 (unhappy) if they have to wait 8 hours and get last choice out of the lineup (for instance, the first person in line if the drawn number was #2). Whether this is realistic or not is up to you.

So what's the expected utility for any given position in line?

As a base case, here's the results for a first come, first served set-up:

Pretty straightforward - in the hypothetical scenario I've invented, showing up 3 hours early to guarantee first selection is much better than showing up right at the cut-off, waiting 5 hours while 599 people buy their tickets first, and then getting last selection out of anyone else in line. Your best bet here is to show up first and get your tickets as soon as possible, hands down.

But what actually happened for Foo Fighters was this:

In this system, the average wait time for anyone still ends up being the same, but there's a much wider variance depending on where the ticket is drawn. The variance in utility in this case is WAY smaller. On average, though, the person who shows up last is better off than the person who shows up first - showing up last is the only position that guarantees you'll move up in line once the lottery starts, an your average time spent in line would only be about two and a half hours. The poor fella who came three hours early is likely to be moved fairly far back, and on average will have to wait in line twice as long as the person who showed up last. Brutal.

So if everyone had paid attention to how the system was going to work, they should have all showed up as close to the lottery draw as possible. Of course that wasn't what happened.

But what if you knew that wasn't what was going to happen? What if, say, you had a friend in line too, and the two of you wanted to know the best positions combined to be to maximize the utility of you two as a team? A bit more of a complicated analysis, but the results look something like this:

The absolutely optimal place to have two people in line is to have one show up somewhat early and try to get in position 300, and one show up right at the end and end up with position 600. By spreading out this much, no matter which ticket is called to be the new front of the line, both members are likely to have no more than an hour and a half of waiting time, with one partner not having had to wait at all beforehand. Splitting it up like this is probably also nicer for the other people in line, because whoever was furthest from the front after the lottery draw could leave the line too!

What's really quite interesting is that the optimal positions aren't always off by 300 - in fact, so long as one member of the pair is in the first 210 positions, it's optimal to have the second to be precisely 390 positions behind. Yay math.

So sure - if you had a really good idea of how the lottery system for ticket pre-sales was going to work, you could game it and do just as well as showing up early to a first come, first served system. This is completely ignoring the fact that, when it comes to lining up for tickets for a show, first come first served is actually a much better idea. I get that Northlands wants people to have an even chance at decent tickets, but people are going to line up early anyway, and they're just penalizing them by being inconsistent with their lottery systems. Keep it simple, and let people decide for themselves how long they want to wait in line.

Friday, October 31, 2014

Women's Inequality in Canada

About 5 months ago the Canadian Centre for Policy Alternatives released a report that compared the "best and worst place to be a woman in Canada." I wasn't a huge fan of the report - in fact, I thought the analysis wasn't too dissimilar from my zombie post, and disagreed with how strict rankings were compared across broad categories. I also thought that calling Edmonton the "worst place to be a woman" in Canada to be a bit of a jump from the findings as reported - Edmonton was instead (by their standards) the lowest-ranked city out of 20 in terms of equality.

Just recently, the World Economic Forum released its very own report on worldwide gender inequality, and I liked it much better. It measures a similar number of quantitative results, but weighs factors appropriately based on statistical measures, and compares them based on scores instead of their rankings between categories. Though the merits of the specific measures used are open to interpretation, I'm satisfied that they're representative of inequality across the world.

These stats were so well prepared and presented, in fact, that I figured Canadian cities deserved the same treatment in their rankings. If we take (approximately*) the same approach as the World Economic Forum, and apply it to Statistics Canada results for our top 20 cities, this is what we get:

Rank City Score
1 Victoria 0.836
2 London 0.817
3 Sherbrooke 0.783
4 Ottawa-Gatineau 0.777
5 Vancouver 0.773
6 Québec 0.772
7 Toronto 0.766
8 Saskatoon 0.756
9 Montréal 0.747
10 Oshawa 0.746
11 Halifax 0.739
12 Winnipeg 0.730
13 Hamilton 0.727
14 St. Catharines-Niagara 0.726
15 St. John's 0.720
16 Kitchener-Cambridge-Waterloo 0.713
17 Regina 0.711
18 Windsor 0.708
19 Calgary 0.693
20 Edmonton 0.692

So there's good news and bad news here. Good news: some cities are pretty much in the same rank as the CCPA study (especially around the bottom of the list). Bad news: some cities moved over 10 spots (London and St. John's basically traded places). In general there's a weak correlation between the two analyses.

The World Economic Forum model looks at four major categories: Economic Participation, Health and Survival, Educational Attainment, and Political Empowerment, all weighted the same. Within each category are up to 5 differently weighted sub-categories - statistical measures that are typically converted into ratios of female:male success, with 1.00 being perfect equality, and 0.00 being complete inequality. Weights within each category are distributed based on the overall variance of that measurement, so that a 1% change within one sub-category is worth the same as a 1% change in another.

There are a couple of advantages to using the World Economic Forum model when looking at Canadian cities. Comparing inequality scores to each other allows for a better overall picture of how the city is doing, compared to a strict rank of cities between each other. Also, by following an internationally accepted standard, we can compare these numbers directly to the results of other countries to get a better idea of exactly how good or bad any given city is.

Some fun findings:

Victoria, the best city in this ranking system, has an inequality index of about 0.836, which indicates that it is more or less as equal as the entirety of Norway. It mostly got this due to being very strong across all categories, but particularly for having the closest to balanced city council participation out of any city in the country.

Montréal was approximately equal in terms of female-to-male equality to Canada on average. Much like the country on average, women in Montréal do just as well or better than men in terms of health and education, but are still trailing behind from an economic point of view and are drastically far behind in terms of political representation.

Edmonton, sadly, still has a lot of ground to cover and I'm afraid there's now way of looking at the stats that doesn't rank it last in terms of equality. Our score puts us approximately equal to Russia in terms of inequality.

Edmonton suffers mostly from having had very little female political representation, as well as significantly fewer women working than men, while earning much less than they do.

None of this is saying that Edmonton is particularly bad place to be a woman - certainly I'm sure that most women feel safer in Edmonton than they would in Russia, and the standard of living is likely much better on average. It is simply the case that Edmonton men have it as much easier than Edmonton women do as Russian men to Russian women.

As a country in general, we still have a long ways to go, and I'm certain Canada can climb from 20th in the world. But that change starts here in our cities, and an analysis like this is (in my mind) much more useful for telling us where we stand than the report by the CCPA five months ago.

*Minor changes from WEF report include:
-Political Empowerment: "Head of state" was changed to "mayor", ministerial positions were combined with members of parliament and replaced with current members of council.
-Wage equality for equal work data wasn't available for cities, so it was combined with estimated earned income.

Spreadsheet available upon request

Health 1, 2
Economy 1, 2, 3
Political: Individual city websites

Thursday, October 2, 2014

McDonald's Monopoly Stats

Man, was I ever excited when I saw that McDonald's Monopoly was back this year! I had a blast looking at the Roll Up the Rim stats last year, and hoped I could do the same for Monopoly this year.

Then I was pretty disappointed to realize that Business Insider had done their own analysis. I was all set to just read theirs contently, until I realized that they just copied and pasted their same article from the year before (hint: the prizes changed this year, dummies). So yay - I get to do my own!

(By the way, a less fun breakdown of the stats is done in the official McDonald's rules, so feel free to check my sources as we go along).

So how this all works is that whenever you buy certain food items (like a medium fountain drink, medium coffee, Big Mac, etc), you win two game stamps. Stamps either have a property, an instant win for food, or an instant win for some sort of other fancy prize. The breakdown for how this looks is approximately this:

A total of 1,303,683,256 stamps were printed. McDonalds' claim that one in four purchases will result in an instant win is bang on then, which is nice. The reasons these numbers may not line up 100% (unfortunately) is that some of their prizes are listed for the U.S. only, and they haven't indicated the exact distribution for Canada.

So one quarter of the time you buy anything, you'll win something. That's kinda nice. I mean, chances are it'll be medium fries (they're 50.2% of all food items, after all), but you could always hold out for that rare Royale with Cheese (you have a 2.2% chance any given time you go!).

The other three quarters of the time, you'll get a property, and if you collect all in a property group, you win big! The problem is that McDonald's doesn't distribute their properties in an even manner, instead they distribute them in such a way as to give you hope. Each colour group has a handful of very common properties (typically a 1 in 11 chance of getting them for any purchase), and one property that's very nearly impossible to get.

Bit tough to make out the really valuable prizes...
For instance, McDonald's has printed off approximately 60,670,043 Baltic Avenue stamps, so the chance of getting at least one any time you play (remember, 2 stamps per play) is 9.09%. Those are pretty good odds - it won't take you very long to get yourself a Baltic Avenue stamp. They only printed off 1,000 Mediterranean Avenues though - so your odds of getting one of those any time you play are only 0.00015%. A wee bit tougher - and don't forget, that's only for the cheapest property prize ($50).

Similarly to how I looked at Roll up the Rim, we can take a look at just how many times you'd need to play Monopoly to have a reasonable chance (>50%) of winning any given prize. The average price of eligible McDonald's items ranges from $1.00 (hashbrowns) to $4.49 (Bacon Clubhouse), but I'll use the average of all eligible items as $3.17 for calculations (because who wants to eat millions of hashbrowns?). If we start with the properties, we'd get:

Mediterranean Avenue: Need at least 451,666 plays to have a shot at getting... a $50 gift certificate! In the process you'll likely win about $170,000 worth of prizes, but it'll result in a net loss of $1,300,000. Maybe not worth it.

Vermont Avenue: Need at least 112,955,548 plays, but you just might win gas for a year! Never mind the net loss of $316,000,000.

Virginia Avenue: Better get ready for 90,364,438 hash browns (or equivalent), for your very own chance at $5,000! Fortunately, by that point you'll likely have had 4 million each of the other two properties for that group, so you're good to go!

Tennessee Avenue: Pretty easy with only 1,895,651 plays needed, and you just might get a Samsung Galaxy! Fun fact: the caloric value of 1,895,651 hashbrowns is enough to feed you for about 415 years in a row.

Kentucky Avenue: Enjoy the necessary 22,591,110 plays you're likely going to need to win your very own 5-night Delta Vacation for Two! If you got that in medium fountain drinks, it would fill five and a half Olympic swimming pools (everyone's favorite measurement of volume).

Ventnor Avenue: Another relatively easy one - only about 6,024,296 plays for a 50% shot at nabbing a Beaches Resort Vacation for your family! If you bought all your plays with bacon clubhouse sandwiches, that would only cost you about $27,049,089, which is actually a pretty good deal (sarcasm).

Pennsylvania Avenue: A bit tougher, but it'll take about 225,911,096 tries for a reasonable chance at a trip in a Cessna private jet! You might think that's tricky, but wait until you put effort towards:

Boardwalk: Approximately 451,822,158 plays needed, but you'd have a glorious chance at nabbing A MILLION DOLLARS $817,572*! Even at the lowest rate of $1.00 per play for hashbrowns, this effort would require your Monopoly opponents to land on your Boardwalk 225,911 times (in original Monopoly, assuming you had a hotel there).

(* Also - the million dollar prize is misleading. Because it's paid out in $50,000/year installments over 20 years, the present value of the money is much less than a million dollars once you take inflation into account.)

Adding everything up, the total value of all prizes offered in Monopoly is $488,423,499.28. With a total of 651,841,628 plays, that means that the expected return of any given play is about $0.75. This may be pretty reasonable if you're buying $1 hashbrowns (get a hashbrown, plus an expected loss of only about 22 cents), but at an average eligible food item cost of $3.17, the loss to McDonald's on each purchase is about $2.42 or 64%, giving it a worse house edge than Lotto 6/49.

So, much like Tim Horton's Roll Up the Rim (or any large promotions, for that matter), I wouldn't recommend McDonald's Monopoly from an investment point of view. If you're already out there gobbling up hashbrowns, carry on though!