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May 13th, 2019 (Permalink)

Still No Vacancy at the Hilbert Hotel

In our last visit to the Hilbert Hotel*, we saw that even though the hotel was full it could still provide a room to a new customer. That's because the hotel has an infinite number of rooms: room 1, room 2, room 3, etc., and there is no highest room number.

The Hilbert Company, Inc., that owns the hotel also owns a railroad line. Mr. Hilbert, the president of the company, decided that a special train was needed to supply an infinite number of guests for his hotel. The railway line uses the same patented Transfinite™ technology as the hotel, so that the Hilbert train can hold an infinitude of passengers. Since it can carry an infinite number of passengers the train is, of course, infinitely long, as is the track that it runs on.

This morning, a train arrived at the small station next to the hotel carrying a full load, but the "No Vacancy" light at the hotel was lit. Now, if the train had been carrying only a finite number of passengers, there would have been no problem. The clerk on duty at the desk knew how to accommodate finite numbers of new guests: supposing that there were n newcomers, all he had to do was move every current guest in room m into room n + m. In other words, the guest in room 1 would be moved to room n + 1, the guest in room 2 into room n + 2, the one in room 3 to room n + 3, and so on. This process would leave the first n rooms in the hotel empty.

However, the clerk had never dealt with an infinite number of new guests. He couldn't very well move the current guest in room m into room m + ∞, since there is no such room! Not knowing what to do with an infinite number of impatient new guests, the clerk called the manager for help.

"No problem!" replied the manager, and he explained to the new clerk how to make an infinite number of new rooms available in the hotel when it was full.

What did the manager tell the clerk?

Solution


*See: No Vacancy at the Hilbert Hotel, 4/29/2019


May 2nd, 2019 (Permalink)

A News Weak Poll

The news media have to report news 24 hours a day even when there is no news, so why not manufacture new "news"1 by sponsoring a public opinion poll? The presidential election is still more than a year and a half away but polls about it are already being conducted. Trying to predict the nominees now, let alone the eventual winner, is likely to be about as accurate as predicting what the weather will be the same time next year2.

I don't suppose that Newsweek magazine is worse about promoting such polls than other news media, but I can't resist punning on its name. Here's a recent Newsweek headline:

Is Joe Biden Unbeatable?
Ex-Vice President Opens Huge Poll Lead Over Bernie Sanders―
but There's a Catch3

The answer to the headline question is, of course, "no", as it usually is4, and the "catch" is that Biden's lead is too early to mean much. Despite its title, the body of this article makes this point:

Former Vice President Joe Biden opened up a 14-point lead over his main rival for the Democratic presidential nomination, Bernie Sanders, after Biden officially entered the race last week―but there was a catch. Other polling suggested there was a substantial number of undecided voters waiting to be won over by the various campaigns. Biden’s early advantage…will likely soften over time as other candidates come to the fore in the race. … Lee Miringoff, director of the Marist College Institute for Public Opinion, told Newsweek it is still way too early to crown anyone as a certainty to win. "The political graveyards are full with candidates who got off to an early lead but failed to win the nomination…." Cornell Belcher, a former Democratic National Committee pollster…described public polls as “completely meaningless,” particularly so early in the primary process.

It's nice when a news article hyping a poll tells you that it's junk.


Update (5/11/2019): In more junk news about polls, we're told:

The latest bit of bad news for Beto [O'Rourke] comes in the form of a Monmouth poll of New Hampshire primary voters. He's tied for sixth place, along with Sens. Cory Booker (D., N.J.) and Amy Klobuchar (D., Minn.), winning the support of just 2 percent of poll respondents, which is getting dangerously close to "statistically insignificant" territory. It's also not much better than truly insignificant candidates such as Rep. Tim Ryan (D., Ohio), former Gov. John Hickenlooper (D., Colo.), and random Silicon Valley bro Andrew Yang, who are all polling at 1 percent.5

Speaking of statistical significance and insignificance, what is the margin of error (MoE) for this poll? The report quoted above does not tell us. It used to be standard practice for American newspapers to report certain basic facts about polls, including the MoE, even though this information was sometimes relegated to a separate paragraph at the end of the article.

In this case, if you want to know the MoE, you have to go to the Monmouth University Polling Institute's report6. There, you can find out that the MoE for this question is ±5.1 percentage points, which is a larger than usual MoE because only democrats were polled.

To say that something is "statistically significant" simply means that it is not likely to be the result of random chance7. The MoE is a way of indicating how large a difference between statistics must be to be unlikely to be the result of chance. In this case, the difference between two candidates needs to be greater than 5.1 percentage points to be statistically significant.

Logically, "statistically insignificant" must mean that such a difference may well be the result of random chance. Given its MoE, any difference between two candidates that's less than five percentage points is a statistically insignificant difference. So, I'm not sure what the author of the quoted article means in saying that candidates polling only 1% are "truly insignificant". If those polling at 1% are already in "'statistically insignificant' territory", then O'Rourke at 2% is in there with them.

The only stand-outs in the poll are Joe Biden, who is the front-runner at 36%, and Bernie Sanders, who at 18% is a distant second. However, all of the other candidates are grouped together in a pack in the single digits, with Pete Buttigieg leading the pack at 9%. While slightly greater than the MoE, the difference between O'Rourke and Buttigieg is not significant8.

However, as mentioned in the original entry above, it's so early in the race that there's plenty of time for O'Rourke to gain ground, and just as much time for Biden to lose it. The New Hampshire primary, which this poll is supposed to be forecasting, isn't until February of next year9.


Notes:

  1. See: Junk Headline, 4/22/2019.
  2. Stephanie Slade, "I'm Betting the 2020 Election Polls Are Junk", Reason, 4/4/2019.
  3. Shane Croucher, "Is Joe Biden Unbeatable? Ex-Vice President Opens Huge Poll Lead Over Bernie Sanders―but There's a Catch", Newsweek, 4/30/2019.
  4. This is an old journalistic rule of thumb, namely, that the answer to any headline in the form of a question is: "no".
  5. Andrew Stiles, "Beto O’Rourke Approaching ‘Statistically Insignificant’ Territory in New Hampshire Poll", The Washington Free Beacon, 5/9/2019.
  6. "Trump Bigger Factor than Obama for 2020 Dem Primary Voters", Monmouth University Polling Institute, 5/9/2019.
  7. See: Victor Cohn, News & Numbers: A Guide to Reporting Statistical Claims and Controversies in Health and Other Fields (1989), p. 176.
  8. I used the following calculator to figure this: "Ballot Lead Calculator", American Research Group, Inc., accessed: 5/11/2019.
  9. See: Jennifer Rubin, "What’s happening in New Hampshire?", The Washington Post, 5/9/2019. Rubin discusses the same poll as that discussed above and begins by noting: "Current polls for the 2020 Democratic primary aren’t predictive."

Solution to Still No Vacancy at the Hilbert Hotel: The manager told the clerk to have the current guest in room n move into room 2n. So, the guest in room 1 would move into room 2, the one in room 2 into room 4, those in room 3 to room 6, and so on. In other words, all of the current guests would move into the even-numbered rooms. There are, in fact, just as many even numbers as there are counting numbers, so this would accommodate the infinite number of current guests. Obviously, this process would leave the odd-numbered rooms empty and, just as there are an infinite number of even-numbered rooms, there are an equal number of odd-numbered ones. Therefore, the new arrivals from the infinite train could be accommodated in the odd-numbered rooms of the hotel.

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