Thursday, June 23, 2016

Snapshots 2

Yesterday I posted some more snapshots to my Google+ profile. I figured I'd start using that outlet for pictures or little notes, while continuing to post blog-post-type things on this blog.

Anyway here's a link to the snapshots, if you want to see them—art and artifacts, buildings, street scenes, airports and hotels. A few of the pics in smaller format below.












 




Monday, June 13, 2016

Solution to Clock Puzzle

On a wall clock, the minute hand (black) and the sweep second hand (red) are the same length, while the hour hand (grey) is shorter. If the clock starts at noon, then at approximately what time after that do the tips of the three hands first become collinear?

Answer:

During the first minute, the hour hand will barely move, and the minute hand will move a bit, so there arises a bit of angular separation between them. The line through their tips intersects the far edge of the clock somewhere to the left of the 6. Thus, there will be a time, somewhat after 30 seconds have passed, when the tips of the clock hands are collinear. Here's a picture of that moment:



If we take the hour hand to be three-fourths as long as the other hands, as suggested in the previous post, then we can give a more precise answer: collinearity first occurs after 33.6 seconds have passed.

Between noon and midnight, there are 708 different moments when the tips of the three hands are collinear.

(This count does not include the trivial instances of collinearity that occur when the minute hand and second hand coincide—there are 708 of those also.)

Here's a movie that animates all 708 of the moments when the hand-tips are nontrivially collinear.  



If you want to know how I found these 708 cases, click here. (I think this is probably the first time I've ever factored a 1,438th-degree polynomial!)

Sunday, June 5, 2016

Clock Puzzle


On this wall clock, the minute hand (black) and the sweep second hand (red) are the same length, while the hour hand (grey) is shorter. At the moment of time shown, the tips of the three hands form a triangle.

Question: If the clock starts at noon, then at what time after that do the tips of the three hands first become collinear?

An approximate answer in the form of a sketch is fine. If you wish to approach the problem quantitatively, then let's agree that the hour hand is three-fourths as long as the other two hands.

Friday, May 6, 2016

Will the Senate Flip?

UPDATE June 20, 2016

In the time since my original post below, one rating has changed from "likely D" to "solid D." Here's an updated chart.



I see that Sam Wang is publishing his model now, so I don't need to update this chart again.


Original post: 

The Cook Political Report rates the competitiveness of Senate races using terms like "solid," "tossup," and so on. Currently,

  • Of the 10 Democratic seats in play, 7 are solid D, 1 is likely D, 1 leans D, and 1 is a tossup.
  • Of the 24 Republican seats in play, 11 are solid R, 6 are likely R, 1 leans R, and 6 are tossups.

The Democrats have fewer seats in play, and the Democratic seats are generally safer. Obviously this situation favors the Democrats. Is it enough to flip the Senate? Currently, the Republican majority is 54–46 (awarding the two Independent senators to the Democrats, since that's who they caucus with).

For the sake of building a toy model, I assigned probabilities to the Cook Report's verbal ratings as follows:

  • For a tossup race, I assigned both outcomes equal probability ½.
  • For leaning seats, I assigned probability ⅔ to the outcome matching the lean.
  • For likely seats, I assigned probability 0.85 to the likely outcome.
  • For solid seats, I assigned probability 1 to the indicated outcome.

These parameters aren't based on any firsthand knowledge of the races in question, they're just ballpark figures.

With probabilities assigned, it is easy to simulate the election many times on a computer. Assuming no coding errors (I did this on my lap during jury duty today), we obtain a probability around 24% of the Senate flipping to a majority for the Democrats.

Also, the probability of a fifty-fifty split is 20%, in which case the Vice President delivers a majority for whichever party wins the Presidential election.

Here is the frequency distribution of outcomes based on a million runs of the election. The blue bars represent flipping scenarios, and you can check that their fractions add up to 24%.




Now, don't take this analysis to the bank or anything. The Cook data are bound to change as time goes on, and on top of that, the probabilities that I assigned to Cook's verbal ratings are somewhat arbitrary. For example, if "likely" means probability 0.75 (instead of 0.85), then the chances of a flip increase from 24% to 34%. Intuitively, more volatility is bad for Republicans because they have more seats in play. Conversely, less volatility favors them; if "likely" is 0.95 and "lean" is 0.9, then the chances of a flip are only 12%.

Tuesday, May 3, 2016

Sam Wang

Tonight Donald Trump appears to have locked up the Republican nomination. If you read Sam Wang, this news has been a long time coming. Here's what the Bayesian analyst neuroscientist said back in February:
...it would take a seismic event to change the trajectory for either Hillary Clinton or Donald Trump.
Media figures are pressured to fill a daily “news hole.” I am not under this pressure. Since I don’t perceive the coming months as being all that suspenseful, I’m tempted to sign off on Presidential coverage until June!
...My guess is that it’s time to get used to these two particular faces between now and November.
This was back when Rubio was still competing for the nomination. I have long believed that Trump and Clinton would become the nominees, and that isn't because of my own perspicacity—it's because, as a Bayesian myself, my practice is basically to decide to believe whatever it is that Sam at the time believes, and to believe in it just as strongly or as weakly as he does.

Update 5/7/2016: Quelle horreur: Sam says he's a closet frequentist! Well, that's OK, we all are sometimes. My toy model of the Senate race from yesterday is just frequencies too.

Saturday, April 30, 2016

Welcome to Oklahomainewmexiconnecticutah!

1. Which state name is last alphabetically?

Wyoming

2. Which state name ends in three vowels?

Hawaii

3. Which state names begin and end with the same letter? (4)

Alabama, Alaska, Arizona, Ohio

4. Which two state names end with the letters LAND?

Maryland, Rhode Island

5. Which state names have a double letter? (9)

Connecticut, Hawaii, Illinois, Massachusetts, Minnesota, Mississippi, Missouri, Pennsylvania, Tennessee

6. True or False? Every letter of the alphabet is needed to write the names of the fifty states.

False: You don't need Q.

7. Which state names could be spelled out very nicely in capital letters using toothpicks because none of the letters are curved? (2)

HAWAII, MAINE

8. Which state name has the fewest syllables?

Maine

9. Which one-word state name has the most syllables?

Louisiana has five syllables—unless you live there!

10. Do more state names end with a vowel or end with a consonant? (Take bets first!)

More state names end with a vowel.

11. Do more state names begin with a vowel or begin with a consonant?

More state names begin with a consonant.

12. One day, the citizens of Florida and the citizens of Idaho noticed that the last part of "Florida" is the first part of "Idaho." So they voted to become one state, Floridaho! Can you think of any other states that could vote to unite because of matching first and last parts?

Looking only for matching parts that have two or more letters, I found these possibilities:

Vermont, Montana → Vermontana
Wisconsin, Indiana → Wisconsindiana
Alabama, Maryland → Alabamaryland
Ohio, Iowa, Washington → Ohiowashington
Oklahoma, Maine, New Mexico, Connecticut, Utah → Oklahomainewmexiconnecticutah

13. I once drove a car through three consecutive states all beginning with the same letter. What were the states?

Iowa, Illinois, Indiana

14. If you want to leave this state by crossing the border into the state that immediately follows it in alphabetical order, you can do that - but only if you're on a boat. What state is it?

Michigan

15. The only two states that have this letter in their name also happen to share a border. What letter is it?

X

16. Starting out from this state, I can drive due north, or due east, and in either case enter a state that has no letters in common with the state where I started. In what state did I start? (2 possible answers)

Kentucky: drive north into Ohio along Interstate I-71, and drive east into Ohio along Route 10 in Greenup County.

Utah: drive north into Wyoming along Pigeon Canyon Road in Daggett County, and drive east into Wyoming along Interstate I-80.


UPDATE 5/1: Changed #16 to correct an error.