I have a question from an example problem in a book I'm reading and am trying to better understand what the author is doing. The example goes something like this:
Suppose you have 100 widgets and they are made by 5 production lines, with each production line producing 20% of the total widgets, i.e each production line makes 20 widgets.
Then suppose the error rate for each production line to produce a bad widget is 2% for all the production lines, except for production line one which has a error rate of 5%.
Draw 3 widgets uniformly at random from the 100 widgets without replacement. Define event A as one of the three widgets being defective. Define event B as: the event that a widget was drawn from production line one.
Then what is P(A|B)?
The book calculates P(A|B) = 3(.05)(.95)$^2$.
But if we consider a simple situation where we have 100 balls and 5 of them are bad, then we have the same percentages and we have:
100*99*98 possible combinations of balls in our 3 ball sample
3(5*95*95) possible draws where we have only one bad ball out of 3.
Then our probability is: 3(5*(95)$^2$) / 100 * 99 * 98 = 135375/970200 = .1395330
But the book yields a probability of: 3(.05)(.95)$^2$ = .135375
Which is correct? Is the book implicitly using "with replacement"?