Probability - Binominal distribution

[MarcMTL]

1. The problem statement, all variables and given/known data
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?


2. Relevant equations



3. 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..

[verty]

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).