Spin again or not
A revolver has a cylinder of chambers. Two bullets are loaded in adjacent chambers, the cylinder is spun, and the trigger is pulled once: it clicks on an empty chamber. The trigger will now be pulled a second time. You may either pull again directly (the cylinder advances to the next chamber) or spin the cylinder first.
Choose the better option. What is the probability that the second pull is also empty?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Label the chambers to around the cylinder with bullets in and . Which empty chambers could the first pull have been on, and which chamber comes next for each?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Spinning resets the odds to the unconditional ones. Not spinning uses the information from the first click: the next chamber depends on where you are, and adjacent bullets make most positions safe.
Option 1: spin. A fresh spin lands on an empty chamber with probability .
Option 2: do not spin. Put the bullets in chambers and , with the cylinder advancing . Given the first click, the first pull was on chamber , , or , each with probability . The next chambers are , , and . Only the last one holds a bullet. So the second pull is empty with probability .
Answer. Do not spin: .
Variant. If the two bullets were placed in random, not necessarily adjacent, chambers, the answer changes. Working it out is a good exercise: the next chamber after an empty one is then empty with probability , which is less than , so you should spin.