Successive Probability and Chance

Continuing with the "math and snowboarding" theme from yesterday, I found that sierrasnowboard.com has a daily snowboard giveaway. The way it works is that if you want to enter, you sign onto their giveaway chat room. They guarentee to give away at least 1 board, but for every 500 people that are in the chat room, they will give away another board. So say for example 687 people show up. They will give away 2 snowboards.

Looking through their "winners" history, it seems that they only give away 1 per day, with the occasional 2 given away. That means that the winners had at worst a 1 in 499 chance to win a snowboard. That seems very low, doesn't it?

This intrigued me a bit and so I wanted to develop out the successive probability of winning a snowboard through sequential entry attempts.

We all understand chance rather well. If there are 500 people in a room and 1 person can win something, then every individual has a 1-in-500 chance of winning. However, what most people assume is that if the event is repeated the number of times of the chance that it is guaranteed that that chance will occur. For our 1-in-500 chance, most people would assume that if the event was repeated 500 times that there is a 100% chance that everyone would win at least once. A small bit of thought reveals this isn't the case.

The reason why is that when people think that they are assuming the event chance is derived by simply adding up all the individual chances together:

(1/500) + (1/500) + (1/500) ...

In reality it's actually more complex than that. We can determine our successive probability by taking the chance that something will not happen and multiplying it against itself. This allows us to progressively determine a smaller and smaller chance that something is likely not to happen. We can then take that number and subtract it from 1 to give us the % that after x successive attempts it will happen:

1 - ( (499/500) * (499/500) * (499/500) )

Lots of math; I know. Anyways, with this formula we've determined that with a 1-in-500 chance, if we were to do something 3 times, we would have a 0.6% chance of it likely happening.

Now that we can calculate successive probability, let's see if we can put a realistic picture on the track record of the SierraSnowboard.com give away. Sierra keeps a tally of the last 2 months worth of wins on their site. Using these numbers, I averaged that approximately 2.12 boards are given away per day. Using this figure, we can infer that there are about 560 people present for each give away. This lends us a 2.12-in-560 chance of winning a snowboard per day. Using this we can determine our successive probability for winning...

1 - (557.88/560)^tries

Attempts	Likelihood of Winning
1 days	0.38%
5 days	1.88%
20 days	7.31%
50 days	17.27%
100 days	31.57%
150 days	43.39%
200 days	53.17%
260* days	62.70%
520 days	86.09%
2322 days	99.99%

So.... it would take about 9 years to "almost guarantee" that you would win one. As it turns out also, the increase in chance to win has a high amount of degradation as the chance itself gets higher. Here's a chart demonstrating this. You can see that as your chance increases, the successive step of increase really slopes off.



* -The event only runs on weekdays, so in a given year (52 weeks), there are a total of 260 weekdays. The event most likely is not conducted during holidays, so the true measure of a year would be lower.