(Part 1 is here – these notes are to assist me, rather than contain any real news!)
So, if the probability that an event will happen to a single entity in a unit of time is and the probability it will not happen is , what is the probability that a large number of events, , will take place?
Considering the Uranium-235 example again, lets say there are a very large number, of atoms: how many will decay and emit an -particle in a given second?
Well we know what the mean number of decays we should expect is: . But this is a random process and so will not always deliver that result, instead there will be a distribution of results around this mean.
What does the this distribution look like – ie., what is its probability mass function (pmf)?
For exactly decays let’s call this .
To show where the pmf comes from, let’s look at a much simpler example – tossing a coin four times and seeing if we get exactly one head, ie .
One way we could get this is like this: HTTT. The probability of that happening is . But that is not the only way exactly one head could be delivered: obviously there are four ways: HTTT, THTT, TTHT, TTTH and so the probability of exactly one head is .
(For two heads we have six ways of arranging the outcome: HHTT, HTHT, HTTH, THHT, THTH, TTHH and so the probability is . For three heads the probability is the same as three tails (ie for one head), and the probabilities for all heads and all tails are both . Cumulatively this covers all the possibilities and adds up to .)
The generalisation of this gives us a pmf thus: , where is the binomial coefficient and can be spoken as “N choose k” and is the number of ways of distributing successes from trials.
There are approximately Uranium atoms in a gramme of the substance and calculating factorals of such large numbers efficiently requires an awful lot of computing power – my GNU calculator has been at it for some time now, maxing out one CPU on this box for the last 14 minutes, so I guess I am going to have to pass on my hopes of showing you some of the odds.
- The binomial distribution, part 1 (cartesianproduct.wordpress.com)
- Power Sum problem (etidhor.wordpress.com)
- The price of a messy codebase: No LaTeX on the iPad (vallettaventures.tumblr.com)
- The ‘swst’ package to print statistical results in Sweave (r-bloggers.com)