Q:

Distribution of $10$ unique coins

You have $10$ coins each with a different probability ($p$) of landing heads. $p = [.05,.1,.15,.2,.25,...,.45,.5].$
What is the likelihood that at least $5$ coins land heads? Is there any more efficient way of calculating than doing the grunt work calculation of all the probabilities of the different outcomes?

A:

Typically conditional probability is the most efficient computationally. Here is an example python code for your case:
 def pgtk(ps, k) :
  if k == 0 :
    return 1

  if len(ps) < k:
    return 0

  return ps[0] * pgtk(ps[1:], k-1) + (1-ps[0])*pgtk(ps[1:],k)

 >>>>  pgtk([0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5], 5) 
       0.0974653128125

