I feel like this question should be simple enough, but I seem to be getting the answer wrong all the time:
A merchant receives two batches of fish cakes (I live in Norway, we always use fish cakes) from the same producer but from
two diferent factories. Thus, from the outside the units of fish cake boxes are indistinguishable
from each other. The batches are called A and B, and they have the following characteristics:
Batch A: 100 boxes, whereof 10 with mushy cakes, described as F1.
Batch B: 100 boxes, whereof 5 with mushy cakes, described as F1; and also 10 with foul
smell and taste, described as F2, whereof 1 with both F1 and F2.
Consider the errors mushy cakes and foul smell and taste to be impossible to detect from the
outside, but can only be detected by destroying the inspected box.
a) Draw venn-diagrams for each of the batches A and B, by regarding the number of boxes.
If the merchant would draw a random box from batch B, what is the probability that
he would draw one with the error F1? If he would draw two boxes randomly from batch
B, what is the probability that he would draw one with error F1 only and one with
error F2 only.
What I did so far to solve it:
Drawing the Venn diagrams was straight forward enough, and also calculating the probability F1, which was simply 5/100. However, when I had to calculate the probability of F1 and F2 I got it wrong. This is how I thought:
The possible ways of drawing 2 boxes from a set of 100 is (100! / 2! x 98!). The possible ways of drawing one box with the error F1 is
(5! / 4!), and the possible ways of drawing one box with the error F2 is (10! / 9!), so I thought the answer should be
(5! / 4!) x (10! / 9!) / (100! / 2! x 98!), which is 0,01. However, this is wrong, the answer in the answer key is 0,00727.
Can anyone explain to me what I am doing wrong?