Question text is in black, my answers in blue.
Given the relative probabilities stated, if both Dr. Watson and Mrs. Gibbon report a noise, what probability should Mr. Holmes assign to the event of a burglary?
You aren't told the prior odds -- what you are asked to do in each of the six situations is to compute the likelihood ratio of the evidence with respect to the event that the Suspect is the Perpetrator. For example, L(ID-S | S=P) is the likelihood ratio for the evidence of the Witness identifying the Suspect, with respect to that event. You need one likelihood ratio for the simultaneous lineup, and one for the sequential lineup. Then you need ratios for the other two possible evidence events, the Witness identifying a decoy or the Witness making no identification at all.
I'll try. For each of the four people in the lineup, #1, #2, #3, and #4,
there is a random choice whether the Witness recognizes that person.
This happens, in Exercise 1, with 90% probability if that person is the
Perpetrator and with 20% probability if they are someone else (an innocent
Suspect or a decoy). These four events are independent (actually, conditionally
independent once we know whether S = P or not).
What the Witness tells the police depends on how many of the four people
they recognize. If they recognize exactly one of the four, they identify
them, and the police know whether they have identified the Suspect or a
Decoy. If they recognize none, or more than one, they make no identification
and the police don't know whether it was none or more than one.
Last modified 10 November 2009