I'm trying to understand the following problem and solution from this page Example 1.19. What does the sample space and $A_1 $event look like? Also, how is $P(A_2|A_1)$ (conditional probability) established? It's my understanding conditional probability uses the form $P(A|B) = P(A \cap B) / P(B)$. I don't see that occurring here. Thanks!
In a factory there are 100 units of a certain product, 5 of which are defective. We pick three units from the 100 units at random. What is the probability that none of them are defective?
Let us define $A_i$ as the event that the $i$th chosen unit is not defective, for $i = 1, 2, 3$. We are interested in $P(A_1 \cap A_2 \cap A_3)$.
$P(A_1) = 95/100$
Given that the first chosen item was good, the second item will be chosen from 94 good units and 5 defective units, thus:
$P(A_2|A_1) = 94/99$
Given that the first and second chosen items were okay, the third item will be chosen from 93 good units and 5 defective units, tus:
$P(A_3|A_2, A_1) = 93/98$
Thus:
$P(A_1 \cap A_2 \cap A_3) = P(A_1)P(A_2|A_1)P(A_3|A_2, A_1)$
$ = 95/100 * 94/99 * 93/98 = 0.8560$
