Odds are Good

This is annoying, because it (again) shows that people have an impoverished sense of what probability means.

Because they can't wrap their head around an "event."

The incident:  Michigan lottery "picks" same 4-digit number two days in a row?  What are the odds of THAT happening?  It must be (pick one:  God.  Fraud.  Sign that probability isn't real.  Etc.)

Well, it depends on what you mean by "THAT" in the paragraph above.  Consider:

• 44 US states have lotteries. Let's say they all have a 4-digit game, to keep things simple.
• There is a lottery result every day, all 365 days per year.
• The chances of hitting any given number is 1/10,000  (because 0000 is a possibility, up to 9999, by ones)

 So, if "THAT" is the chance of the Michigan lottery having exactly the same number two days in a row, then THAT is pretty unlikely.  It's 1/10,000 every day, because it's the chance of hitting yesterday's number again today.

But if we are talking about one state lottery somewhere (there was nothing special about it being Michigan, ex ante) on some day in given year (there was nothing special about those two days), then THAT is just the chance one lottery out of 44 picks the same number on consecutive days, out of 365 (since it could happen on the first day, but that would be across years).

If there are 44 4-digit lotteries every day, and the probability of getting a different number in each particular lottery is 9,999/10,000, that means that the probability of duplicate numbers in SOME state (out of 44), on a given day, is .00439.

But we do that 365 times per year.  Since the chance of no duplicates in all 44 states, on a given day, is .9956, the chances of no duplicates for a year is .9956^365 or .2007.

If that's right (and I'm just doing this back-of-the-envelope, so I've probably made a mistake in logic or calculation!), that means that in any given year the chances of a duplicate lottery, two consecutive days the same number, in some state, is about 80%.

Does that sound right?  If you carry out to multiple years, say 5 years, the chances of getting at least one duplicate in at least one state are better than .999.  It will be a little more complicated in real lotteries, because they are not all simple "pick four digits between 0 and 9," but the same sort of logic applies.

With the caveat, again, that I have likely made a mistake.  The question, then, is whether consecutive duplicates are really as common as this calculation implies.  Thoughts?

Example.....  Example.....  Example..... Explanation.

Excellent example...

Lagniappe:  Scott de M suggests an exercise, left to the reader:  Prove that some athlete, somewhere, in some sport, has a jersey number that matches both  his age and number of wins he has played in.

If Only We Had Something Valuable We Could Sell.....

An email from a reader.

Seattle's push for a $15 minimum wage isn't the only economic illiteracy we have here on the left coast. Here's an additional slice of annual economic illiteracy from our local zoo. 

1. The zoo needs money (they're always asking for donations at the gate). 

2. The zoo has something very valuable, where the demand is much higher than the supply. So what does a Seattle zoo do (ha!) with this valuable resource? Do they auction it to the highest bidder(s), perhaps with a dutch auction, thereby maximizing some much needed revenue? No, of course not, this is Seattle! 

Since this special fertilizer is limited, [they are going to sell it, right?  no...]  you have to enter a lottery for the chance to purchase... Sigh. Face palm. Double face palm.

Tullock Mentioned in ASR? Wow.

By the extraordinary G. Rossman, the world's last sensible sociologist.

 His article "Close, But No Cigar: The Bimodal Rewards to Prize-Seeking," actually by Gabriel Rossman and Oliver Schilkea. A link if it lasts. Gated link if it doesn't. Wow!  In the ASR.  Much respect.  The word "Tullock" in the ASR?  The force is strong in this one...

Abstract:

This article examines the economic effects of prizes with implications for the diversity of market positions, especially in cultural fields. Many prizes have three notable features that together yield an emergent reward structure: (1) consumers treat prizes as judgment devices when making purchase decisions, (2) prizes introduce sharp discontinuities between winners and also-rans, and (3) appealing to prize juries requires costly sacrifices of mass audience appeal. When all three conditions obtain, winning a prize is valuable, but seeking it is costly, so trying and failing yields the worst outcome—a logic we characterize as a Tullock lottery. We test the model with analyses of Oscar nominations and Hollywood films from 1985 through 2009. We create an innovative measure of prize-seeking, or “Oscar appeal,” on the basis of similarity to recent nominees in terms of such things as genre, plot keywords, and release date. We then show that Oscar appeal has no effect on profitability. However, this zero-order relationship conceals that returns to strong Oscar appeals are bimodal, with super-normal returns for nominees and large losses for snubs. We then argue that the effect of judgment devices on fields depends on how they structure and refract information.