Roswell in the news. Again.

Here's a fox news story that came out today: Dead Airman's Affidavit: Roswell Aliens Were Real. It discusses a death-bed confession of a person apparently involved in the original story 60 years ago.

I admit and agree that these stories sometimes make one second guess what really happened out in the desert at the start of the cold war. But really, why would aliens who have mastered space travel be coming out her to monitor Earth? Why? Surely close to 100% of what they'd need to know could be gleamed from remote sensing, so why buzz us? If we found primitive life on a planet we would either get down in there and get involved or just put a satellite in orbit to observe. Whatever, the arguments are numerous.

But people will always be fascinated by the possibilities. I don't really doubt there's life out there somewhere. It's almost a statistical certainty. There's a recent article in Scientific American about the odds for life on Mars (slim) and Titan (better) - and those are just in our solar system. But being capable of space travel and actually coming to visit Earth to observe/abduct humans borders on absurd.

Half Pills

Every night I've been taking a half of a pill. They come in a typical little Nature Made bottle. My brain being as it is, each and every night I wonder if I'll tap out an already split pill or if I'll have to break another one in half. So I started thinking about the odds.

Clearly the first day by odds are 0% to pull out a half pill. I break one in half and return the other half to the bottle. On day #2, there is now 1 half pill and 49 whole pills, so my odds are exactly 2% in pulling a half pill. From here the numbers get non-deterministic because it depends on if I pulled the half pill or not. If I did on day #2, my odds drop back to 0% for day #3 since there are no more halves. If I had to split another one on day #2, then on day #3 there are 2 halves and 48 whole pills giving the odds of 4%. While running some simulations might be interesting to get some typical behaviors, the question that kept popping in my head is what do the odds really look like over time?

This is easily computed in a spreadsheet because the odds on the next day depend on the day before. While this isn't applicable to any particular day (since your particular observations depend on your current count of whole and half pills), it might make a pretty picture. Looking at the math, on day #1, there are 0 half pills and 50 whole pills. Given. On day #2 there is 1 half pill and 49 whole pills. Given. On day #3 you have a 2% chance of grabbing that half pill, and if you don't you add to the number of half pills by 1 (a 98% chance). Mathematically then the theoretical number of half pills on day #3 is the number on day #2 (1), plus a 98% chance of adding 1, minus a 2% chance of removing one. Or if p2 = percentage chance of pulling a half on day 2 and h2 = number of halves on day 2, then the number of half pills on day 3 is:

h3 = h2 + (1-p2) - p2

The number of whole pills on day 3 is the number on day 2 minus a 98% chance of pulling another whole one. Using 'w' to denote whole pills we have:

w3 = w2 - (1-p2)

Since we're interested at the percentage on day 3, we compute the quotient of the number of halves divided by the total of these two values, or:

p3 = h3/(h3 + w3)

Since this is too ugly to do in HTML, I'll leave it at that instead of building the whole equation. If you plug in the numbers for day #3, in general you have a 3.92% chance of getting a half pill since there are 1.96 half pills and 48.02 whole pills in the bottle. It's not obvious maybe, but there's a check you can do on the math. Consider that there are 100 total "doses" in the original bottle. After each day, the number of doses drops by 1, whether or not you had to break a pill. Therefor at any given day, mathematically the sum of 2*hN + wN = 101-N where N is the day#. Checking then, on day #3, 2*1.96 + 48.02 = 98 doses. The 101 is there instead of 100 because of where I started counting days (day #1 sees h1=0, w1=50, d=100)

You can write the equations recursively where you replace the '3's above with a subscripted "n" and replace the '2's with a subscripted "n-1". This lends itself nicely to a long column in a spreadsheet, computing the odds from day #1 through day #100. Because of the nature, we don't end up with exactly 1 half pill on day #100 and a 100% chance of getting it. But if we round to integers and step through the values, that is exactly what you will end up with of course.

Now have a look at the graphic for today's blog. This is a chart tracking various values from Day #1 through #100 on the x-axis. Counts of the number of half pills (red line, left y-axis) and the number of whole pills (green line, left y-axis) flow smoothly and as you might expect. The number of whole pills steadily decrease over time, flattening out a bit as the number of halves and wholes get closer past day #50 or so. The number of half pills increases quickly at first, then slows as they become numerous before the odds catch up and they also decrease at an ever increasing rate. The odds of pulling out a half pill (blue line, right y-axis) starts very low but increases continuously. Again, it should be 100% for all real-world cases on day #100, but here it ends at about 87%.

There are some assumptions in this graph:

• The bottle is perfectly well mixed (half pills don't settle at the bottom)
  
• You tap out only a single pill each time
• If you do pull out multiples you randomly pick (which is impossible, so see previous)
  
• You take the pill that you tapped out instead of looking for a half because you're too tired and lazy to break one that night

But I think to illustrate the point, these assumptions are all OK. I'm not trying to simulate human behavior here, but just figure out what my probable odds are of pulling out a half pill. I've verified for myself that after day #45 I'm more likely to pull a half pill while before then I should be prepared to always have to break one.

Lose weight by drinking Scotch

Losing weight, of course, is a matter of burning up more calories than you take in. A calorie, as everyone knows, is defined as 'the amount of heat required to raise the temperature of one gram of water one degree centigrade.'

Let us take a good glass of Scotch and soda. Since a gram of water is pretty close to 1 cc (to make it simple), put in plenty of ice and fill it up to about six or seven ounces, making it, say, 200 cc. Since it contains melting ice, its temperature must be 0° Centigrade (neglecting the temperature-lowering effect of the alcohol, Scotch, and gas).

Sooner or later the body must furnish 7400 calories (200 cc x 37° C) to bring it up to body temperature. Since the calorie-counter books show Scotch as 100 calories per shot, and club soda as 0 calories, we should be able to sit around all day, drinking Scotch and soda, and losing weight like mad.

The above is a direct quote from an old book I have. Sounds awesome, doesn't it?

Clearly, since this isn't the case there's a problem with the problem. Can you figure it out? Hint: it's not in the assumptions or approximations.

Take a minute and see if you can figure this out. You can find the answer here in about the third paragraph down.

As for me, I don't care if it's right or not -- I'm going to give this one a good run for the money

Uunartoq Qeqertoq

The Warming of Greenland. Pretty interesting (and kind of scary) stuff. So much ice has melted, we're finding new islands now. And it's apparently not a freak thing, but becoming a "regular phenomenon" which is the scary part.