Quote: (06-06-2013 11:16 AM)lurker Wrote:
Whenever you're asked to find the probability of something occurring at least once through a set of events with defined probability, the easiest way to do it is to calculate the probability that it never happens and subtract that from 1.
The probability it never happens is the product of all the individual probability events that it doesn't occur. Here, there's a 50% chance any one night is starry, which means there's a 50% chance any one night is not starry. Thus, the probability that both nights are not starry is 0.5 * 0.5 = 0.25. This means the probability that at least one night is starry is 1-0.25 = 0.75.
Likewise, the probability of at least one night in three being starry would be 1- (0.5*0.5*0.5) = 1-0.125 = 0.875.
If he needs to know the probability of exactly X number of nights being starry out of N total, he needs to use the binomial distribution.
This. The other guiding rule here is that for 2 INDEPENDENT events, in order to get the probability that they both happen, you multiply the probabilities together.
Not to confuse the issue, but where this often breaks down is in the real world where it's much harder to measure the degree of dependence (i.e. wind currents on cloud cover, condensation, light pollution, etc.)
Which also spills into an oversimplified understanding of risks and probability in society (see the writings of Benoit Mandelbrot and Nassim Taleb).