NOT a conditional probability problem
Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percent wear a ring and 30 percent wear a necklace. If one of the students is chosen randomly, what is the probability that this student is wearing (a) a ring or a necklace? (b) a ring and a necklace?
A conditional probability problem
All bags entering a research facility are screened. Ninety-seven percent of the bags that contain forbidden material trigger an alarm. Fifteen percent of the bags that do not contain forbidden material also trigger the alarm. If 1 out of every 1,000 bags entering the building contains forbidden material, what is the probability that a bag that triggers the alarm will actually contain forbidden material?
How do we distinguish the two?
For the second problem(before I knew it was a conditional probability problem) I interpreted it as
$A$= triggers the alarm
$F$= contains forbidden material
$P(A \bigcap F)$ =.97
$P(A \bigcap F^C)$ =.15
$P(F)$ = $\frac{1}{1000}$