Source of original question and answers can be found here under "Exercise 1"
http://www.intmath.com/counting-probability/9-mutually-exclusive-events.php
A box contains 100 items of which 4 are defective. Two items are chosen at random from the box. What is the probability of selecting:
(a) 2 defectives if the first item is not replaced;
(c) 1 defective and 1 non-defective if the first item is not replaced?
I am having issues understanding (c). When I did (a), it was a simple dependent events kind of problem, initially there are 4 defectives, you pick one out, and now there are 3 defectives out of 99.
However, when I did (c), I reasoned through it similar to how I reasoned through (a), and my math was:
(4/100) X (96/99).
This was wrong as the answer said we need to account for the other possibility: what if a non-defective was picked first and then a defective. (C) reads to me very similar to (A). The difference is there is an "and", someone told me it is commutative, doesn't matter what comes first, you must account for all possibilities.
But I get my head confused about other "and" probability questions. This one categorized as a mutually exclusive example. But when I think of other AND dependent probability questions I've done, I didn't have to solve for the other possibility. For example: colored marble problems, there are X blue marbles and Y red marbles in a bag, what are the chances of pulling a blue and then a red without replacement. Those questions only require to consider the first scenario, not account for "oh what if I picked a red marble first instead of blue?"
When I read problem (c) it seems that order does matter, the first is defective, not maybe it could be the other way.
Am I reading it wrong, if the problem instead said "the first item as defective and the second as non-defective if the first item is not replaced" would my original answer be correct?
Best Answer
Indeed, the conjunction matters. When reading (and writing) probability problems, the following forms of event criteria should have this standard interpretation: