Failure to Understand the Concept of Randomness: Any Psychologists in the House?

Factual Questions
1

I work in a group home for developmentally disabled adults. One of the residents, call him “Don,” is a man with an IQ of 43. He loves playing Battleship with me every day.

The interesting thing about Don’s Battleship playing is that there’s one aspect the game that, after over a year of playing with him every day, he still does not get. He doesn’t understand the concept of calling random squares across the board to try to find my ships.

For example, Don begins **every game ** by calling A1, then A2, then A3, etc. Interstingly, when he gets a hit (say on A3), he’ll call the next square in sequence (A4) but if that is a miss, then he’ll call the square below (B3), just like I’ve taught him. He’ll call until he gets a miss, just like I’ve taught him, but then he’ll begin sequencing again from the square where he got the miss. For example, if he gets hits on A3, B3, and C3, and a miss on D3, his next calls will be D4, D5, D6, etc. The man simply cannot grasp the concept of picking a square at random and calling it, even after over a year of patient instruction.

I’ve found this an interesting case study, but I’m curious if my findings have been corroborated. Have psychologists observed this in others? Is there a part of the brain that understands randomness that simply isn’t present in the developmentally disabled?

:confused:

2

I think the average person doesn’t understand randomness, let alone the developmentally disabled.

I don’t know how many times I’ve heard someone say, “These drug tests can’t be random; this is the third time I’ve had to take one in two years” or “Well, somebody’s got to win the lottery.”

3

I think that actually shows some inherent UNDERSTANDING of randomness non-disabled people, albeit one that is not mathematically rigourous. People instinctively know what a “random” pattern should look like (or at least will USUALLY look like, the sequence A1,A2,A3,etc is just as likely as a “random” looking one), and will smell a rat if it doesn’t, even if mathematically speaking they are incorrect.

4

Order and sequence are extremely important for the developmentally disabled. That’s why so many sheltered workshops specialize in assembling things. It’s much easier to learn a pattern of A then B then C than it is to understand that you can do A,C and B in any order and still get to the desired result, or even if you run out of C, you can still do A and B until you get more C.

It’s not just the DD, either. My father, a chess master, told me how he lost a tournament one time because his opponent varied from a standard gambit, and in the several moves it took my father to catch up, the opponent had grabbed the advantage.

5

I wonder if maybe the explanation for your Battleship Buddy’s behavior may be an extreme form of the phenomenon where having too many alternatives to choose from can have a paralyzing effect. This book (which I have not read) appears to describe the phenomenon; here and here are a couple of articles that Google gave me about the same sort of thing.

6

As far as understanding randomness, the theory I have often encountered is that we are hard-wired to see patterns, connections, and correlations. Since so few things in nature are truly random and independent, we have trouble grokking situations (like in human-designed games of chance) that are, which may be the origin of the gambler’s fallacy.

7

Perhaps Don’s problem is not so much that he doesn’t understand randomness, but that calling out random cells gives him no advantage or even a disadvantage. With his limited IQ, he may not be capable of remembering past cells that have been called randomly and that can lead to confused play on his part . By calling out cells in some fixed order gives him a way of actually playing the game for someone with his skill set.

8

Except that he understands putting a white peg in a cell that his been called a “miss,” so he doesn’t need to rely on his memory. He seems to grok that part of the game.

9

It depends on your definition of randomness. While any sequence may be equally probable, sequences with simple patterns don’t look “random”. See “Kolmogorov randomness”*.

  • (from Wiktionary) Informally, the property of a string being not longer than any computer program that can produce that string; that is, the property of a string being incompressible.
10

Are you sure that Don would be at a disadvantage against a new opponent? Having called A1, the chances that A2 would contain a ship should be the same as the odds of any other square on the board, except for the fact that you know how he plays and can effectively cluster your ships on the bottom right hand of the board. Perhaps Don counts upon your sense of fair play to not take advantage of his pattern of play.

Once he gets a hit he plays properly, clustering around the square that he made a hit on. If Don were playing roulette and played the numbers 1,2,3,4,5,6 in that order would he be more or less likely to win than if he’d played six random numbers?

11

As a strategy, calling every square is unnecesary, since the smallest ship occupies two squares, so testing every other square will eventually hit every target. There’s also the chance that a larger grid might by pure luck hit one of the smaller ships. So optimum strategy would be a search grid designed to find the largest ships first, sink them, then any smaller ships that got found in the process, then fill in successively smaller holes in the search net.

12

I don’t think this is true. My greatly simplified reasoning: There are only three ways a ship can be on A2 (horizontal left and right, and vertical down), while there are four ways a ship could be on B2 (add vertical up) it seems to me that the squares on the edge would have a lower chance of having a ship, linearly increasing towards the middle of the board until you get as many rows or columns in as the longest ship has squares.

Sorry if this is a hijack.

13

Almost. I think that the best strategy for Battleship is to search with a grid of length 4, because if you don’t find the small ship doing that, then you minimize the number of shots you have to use to reduce the partial grid to one of length 2. But that might not be true. If you go with a grid of 5 and happen to find the 2-ship, then you collapse to a 3-grid more efficiently. I’m going to have to write some Battleship simulators and get back to you.

Not directly related to randomness, but have you considered telling Don the strategy of “pick a cell that’s farthest away from all the other shots that you’ve made”? That may or may not be within his abilities, but it does specifically describe a strategy in a way that “pick a random square” doesn’t. It seems to me like “farthest away” might easily be calculated even without math skills, just by looking for a hole in the pegs. But I have no real understanding of his mental processes, so that’s just a WAG.

14

There is no randomness in Battleship. Each player tries to place his ships in positions the other player won’t expect, and fire his shots where he suspects the other player has chosen to place his ships. This leads players to often try simulating randomness in their choices, but most people are not even very good at simulating randomness. The best strategy, therefore, is to figure out what patterns people tend to produce when they try to simulate randomness. A good Battleship player may even use this strategy by accident, intuitively expecting a pseudorandom arrangement of ships and spreading shots out across the board.

In other words, what Don seems not to have grasped is the erroneous, but widespread, image of a “random” scatter that most people have.

15

Am I correct in thinking that any deliberate Battleship strategy would beat random play half the time?

16

Actually, I think depending on the context, that could show a pretty sophisticated understanding of probability; in particular that it is improbable that one particular ticket picked ahead of time wins the lottery, but not improbable that there is a winning ticket.

17

I wonder if it’s really the concept of randomness that “Don” doesn’t get, or the concept of an arbitrary choice between alternatives that offer no reason at all to prefer any of them.

From my experience the latter is something that I have frequently encountered with people of average or even with above average intelligence.

me: We have installed this new Russian font ::selects font:: Please write something to look how it works on the screen.
Colleague: OK. What should I type?
me: Anything - it doesn’t matter.
Colleague: ::looks like a deer in the headlights::
me: ::sigh:: Is there a Russian sentence that is the equivalent of “Hach welch Zynismus, quäkte Xavers jadegruene Bratpfanne”?
Colleague: ::happily types:: В чащах юга жил бы цитрус? Да, но фальшивый экземпляр!

me: Now you can run the script with any of the files in this directory as argument.
Colleague: So, which one to use?
me: No matter. The content of the file isn’t processed at this point
…
me: ::sigh:: For example, you can use a.csv as argument.

(on preview: this is also the subject of Thudlow Boink’s first post)

18

Seems to me that Don’s strategy is a legitimate one, until his opponent capitalizes on his tendencies, and clusters all his ships in the bottom right quadrant.

But that would lead to a different question, then. :wink:

19

Yeah, randomness is tricky.

For instance, I remember a friend dragging us to the indian casino so we could get the seafood buffet. He liked to play Keno…you pick five numbers out of 1 to 100, the numbers start to come up, and you win if some of your numbers match.

Anyway, the point was that you picked numbers and then watched to see if your numbers matched the random numbers the casino generated. So of course, I picked 1, 2, 3, 4, and 5. And he got upset, because he said there was no way that set of numbers would show up. He understood that the numbers were completely random, but I just couldn’t convince him that if the numbers were random then any set of numbers that I picked was just as likely to win as any other set. So while 23, 48, 15, 60, 75 might SEEM as though it was more likely to be chosen, 1, 2, 3, 4, 5 was really just as likely. He didn’t buy it and to this day is convinced I’m an idiot.

The thing about Battleship though, is that a random arrangement of ships isn’t just as good as any other. You want to avoid situations where your ships touch each other, because when your opponent gets a random hit he’ll have to search around to find the rest of the ship. If a second ship is touching the first ship, he’s very likely to find that second ship while searching for the rest of the first ship.

When humans generate pseudorandom arrangements, they very often don’t allow the results to cluster as much as true random placement does.

Don would do a lot better if he skipped every other square, but if he were playing against a computer that placed its ships truly randomly and didn’t remember Don’s playing style from game to game, always starting from A1 is just as good as any other spot. The trouble is that humans are less likely to put a ship at A1 because it doesn’t seem random to them. Plus, a human opponent could realize that Don always picks A1 first, and thus put all his ships in the bottom right corner.

20

It’s a bit misleading to refer to an optimal exhaustive Battleship search pattern as a “grid”: It ends up looking more like a set of diagonal stripes. Even there, though, you can pick the cells on those stripes in arbitrary order, and which ones are best to use first depend on how you’re guessing your opponent arranged his ships. And as you find ships, you’ll invariably deviate from your original pattern of shots, which will also perturb your “grid”.