text: # Probability - Binominal distribution
  
  MarcMTL
  
  ## Homework Statement
  
  A factory makes glass bottles, and 15.0% of the bottles are considered defective. A defective bottle will break 60.0% of the time when dropped from a controlled height, whereas a non-defective bottle will only break 10.0% of the time in the same conditions.
  
  You pick a single bottle at random and drop it four times in a row. It doesn't break. What is the probability that it is a defective bottle?
  
  ## The Attempt at a Solution
  
  If a defective bottle breaks 60% of the time, I calculated the probability of it surviving four consecutive drops: (1-0.60)4 = 0.0256 (2.56%)
  
  This is where I get doubtful, I'd tend to say that if 20% of the bottles are defective, and we picked a single bottle at random, that the probability that the bottle is defective would be:
  
  0.0256 * 0.20 = 0.00512 (0.512%)
  
  Not the right answer. Also, I feel that the last step was incorrect.
  
  Any help would be greatly appreaciated!
  
  Marc
  
  hgfalling
  Try thinking about the whole population of bottles.
  
  X% are not defective.
  Y% are defective.
  
  Now you pick a bottle at random and drop it four times.
  What can happen?
  
  It could be defective and survive. (a%)
  It could be not-defective and survive (b%)
  It could be defective and break (c%)
  If could be not-defective and break (d%)
  
  Now of course a+b+c+d = 100%. And we know that what actually happened was that our bottle didn't break, so we are actually only looking at (a+b)% of the population.
  
  That should get you started..
  
  Homework Helper
  Another way is...
  
  S: the bottle survived 4 drops
  D: the bottle is defective
  
  Find these probabilities:
  P(S:D), P(S:!D), P(D), P(S).
  
  Then use Bayes Theorem to find P(D:S).