Friday, 26 June 2009

The Power of Probability

I wanted to take a quick detour into probability today, for various reason which I don't really want to divulge.

But in particular I wanted to look at the cumulative probability of multiple independent events. Such as always rolling a 6, or always drawing a red card from a full deck.

It is fairly simple to see that if you want to roll a 6 there is a 1 in 6 or 0.16666... probability of this happening. The same goes for always drawing a red card where the probability is 1 in 2 or 0.5 (or 50%).

But once you start to choose multiple times then you see the the probabilities start to get smaller quite quickly. So the probability of rolling 2 6s, or drawing 2 red cards

  • P(2 6s) = 0.1666 x 0.1666 = 0.16662 = 0.0278 = 1/36 = 2.78%
  • P(2 red) = 0.5 x 0.5 = 0.52 = 0.25 = 25%
And then the probability of three
  • P(3 6s) = 0.16663 = 1/216 = 0.46%
  • P(3 red) = 0.53 = 0.125 = 12.5%
You can see that the dice roll is now quite unlikely less than half a percent and even the high probability drawing a red card from a full deck (which is the same as flipping a coin and getting a head - not sure why I didn't use this as my example) is down to 1/8.

What about if we say drawing 10 cards that are all red, from a full deck
  • P(10 red) = 0.510 = 0.0977%
Which as you can see is really unlikely. So over 99.9% of the time you will not get 10 red cards drawn in a row (or 10 heads on a coin toss).

Of course our examples here uses the same event happening over and over again, but the same math applies when considering events with different probabilities as long as they are independent (which means that one event does not influence any others). As long as the probabilities of the events happening is less than 1 (which would mean that that particular event always happened) then the chances of a series of events happening is more and more unlikely as more and more events are in the series.

One place where this knowledge can be quite useful is in a court case. If for example there are several pieces of evidence that show that the accused could have done the crime. What the defense will try to do is cast doubt on those pieces of evidence, such as by saying that this bloody handprint could have gotten there in a total innocent way. Of course the whole idea of the defense is to cast reasonable doubt as to the accused's guilt. But the more pieces of evidence that there are against the accused then the lesser the probability of them all being circumstantial and the greater the likely hood of the accused being guilty. Even if there is some doubt about each piece of evidence taken together they point to the accused as being guilty, even if the doubt is large such as 50% like we showed above.