[ExI] psi yet again.

[Damien Broderick]

On 6/29/2010 4:02 PM, Mike Dougherty wrote:

> I think this is modified the same way 'predictive' analysis of the
> stock market by machines exacerbates instabilities.  If the rate is
> 1/1000 and there are 3 psychics the effect may be accumulate to
> 1/(1000^3)

NOT what I'm saying.

Blimey, how hard is this to follow?

You buy a Lotto ticket (I use Australian data from 20 years ago, that's 
all I know about) and you cross off 6 of the 45 numbers. (Here I 
understand it's typically 6/44 or 6/49, and there are usually extra 
supplementary numbers that allow for smaller prizes--but let's ignore 
those for the moment).

You mark your first guess. Let us imagine (think of it as a science 
fiction story if that steadies your nerves) that there's one chance in 
1000 that a psi flash will cross your mind, either confirming your guess 
or tempting you to change your guess. You do this five more times, and 
submit your entry.

166 of your friends mark their own entries.

One of those 167 entries will contain one extra correct number than it 
would without psi.

Does this mean (as I suspect John Clark would have us believe, since he 
thinks psi would destroy Lotto) that 1 in 167 bettors will share in the 
major prize? Well, no.

Here is the mean chance expectation of the distribution of guesses in 
the Tattslotto game I studied:

All 6 correct	1.223 by 10^-7
5 of 6		2.726 by 10^-5
4 of 6		1.365 by 10^-3

Imagine an idealized 6/45 Lotto game in which exactly 8,145,060 people 
enter, but each person chooses a different pattern of six numbers.  Note 
that LOW-scoring Lotto entries win NO prizes, and are therefore 
invisible. It's simple to calculate how many people should fail to pick 
any winning numbers, how many will get only one right, and so on.

It's all too easy for players to slip to the bottom of the probability 
bucket. There's a little less than even money odds of choosing only a 
single winning number. Indeed, it turns out that it's slightly easier to 
guess one right than none at all, for the chances of getting all your 
guesses wrong drops a little, to two in five!

Number right		 Mean chance expectation	Probability - one chance in...
out of 6			

6				    1					8,145,060.00
5				   234				  34,807.95
4				  11,115				   732.80
3				 182,780				    44.56
2				1,233,765				6.60
1				3,454,542				 2.36
0				3,262,623				 2.50	

Due to the pyramid structure shown above, the great majority of players 
will select either 0, 1 or 2 of the 6 winning numbers. Forty percent 
will be wrong on every guess; another 42.4 percent will get only one of 
the 6 right; and a little over 15 percent will identify two of the 6 
right. Added together, these worst outcomes make up nearly 98 percent of 
all bets.

And there's very little chance that anyone in those categories could 
jump up to the top category when the chance of getting an extra guess 
right is only 1 in 1000, even if 10 million or even 100 million entries 
are made in a draw.

I leave the rest for your exploration and entertainment.

Damien Broderick