Lottery

aka The Power of Personal Testimony

Q: Let's say there's a 3-digit lottery for one million dollars every week, in a country of 3 million people. I set up a fortune telling web site called lucky-number.com and give out lucky numbers. It's free, but if you strike it a second time, you have to give me a 20% cut on your second winning. That's just 200,000 out of your 2 million. Fair? Do you think I can make any money?


Disclaimer : I'm sorry if I did my math wrong!

A: Firstly, given that numbers are given out evenly, every week 1 out of every 1,000  will strike lottery. Thus the chance of anyone striking using the given number is 0.001. Let's call these numbers 1,000 and 0.001 n=1000 and p=0.001.

We always give people a second chance when they don't do well. That is, that they need to lose twice in a row to stop buying - meaning unless they lose the first 2 times consecutively, they will end up trying at least 3 times. So we first consider the following scenarios:

(win, win, win) 
and believe that the site is true
 The possibility of striking (win, win, win) is p³, which means 1 out of 1,000,000,000 people will get this. since we have only 3 million people, let's assume this does not happen.

(win, lose, win) or (win win, lose) or (lose, win, win) 
and conclude the site is mostly right 
People who experienced (win, lose, win) and (win, win, lose) and (lose, wine, win) will conclude that the site works most of the time. So the probability of people thinking that it mostly works is 3(p²(1-p)), meaning 1 out of 333,667 people will feel that the site is useful. That is 8 people in 3 million!This means we would get either 8 times our 20% cut which is $1.6 million by the 3rd draw!

(win, lose, lose) or (lose, win, lose) 
and conclude that the site changed their odds from 1 out of 1,000 to 1 out of 3.
The probability of this is 2((1-p)²p), meaning 1 out of 501 people will come to this conclusion. For a population of 3 million, this is 5988 people!

So now we have 8 people convinced that the site is mostly true, and 5988 people who are not totally convinced, but "feel good" about the site.

Now, let's say each person is directly connected to 10 other people, and they tell these 10 people about their lottery. And their up to their friend's friend trust what they said - beyond that it's too far away to be trustworthy. Let's call this number c=10.

So many people can these 5996 people reach?

So we have friends who number 5996c and friends of friends who number 5996c², and if we include the initial 5996, this is a total of 665,556 people. In a community of 3 million, this is 22%. That means 1 out of 5 will feel good about the site! If you're connected to 10 people, it means you will know 1 to 2 persons who feel good about the site!


Let's say these 665,556 people decide to buy a 4th time.

(win, lose, lose, win) or (lose, win, lose, win)
Those who struck 1 time the first time, would have to pay us if they strike this time - the 5988 people who stuck (win, lose, lose) or (lose, win, lose). What's the odds?  (2((1-p)²p))p, that is one in 501,001. Bingo. We have another winner. Add $200,000

Unfortunately this doesn't grow our "feel good" crowd. But still, with some optimism, we would have collected $1.8 million by the 4th draw, all by giving out random numbers!

Moral of the story
1. Personal experience does not constitute statistics
2. Don't take information second hand.
3. Almost All isn't anywhere near All. When you listen to a testimony first hand, don't skip the details cos that's where the devil lives!