A standard conditional-probability question, and a good one because the intuitive answer is wrong in an instructive way.
The setup
A six-chamber revolver is loaded with two bullets in adjacent chambers. The cylinder is spun, the trigger pulled, and it clicks - empty. You must pull once more. Should you spin again first, or just pull?
If you spin
The cylinder is randomised. Two of six chambers are loaded, so the probability of being shot is 2/6 = 1/3.
If you do not spin
This is where the adjacency matters.
Label the chambers 1 to 6 with bullets in 5 and 6. You survived the first pull, so you were on one of the four empty chambers: 1, 2, 3 or 4, each equally likely.
The next chamber is the following one:
- From 1, next is 2 - empty
- From 2, next is 3 - empty
- From 3, next is 4 - empty
- From 4, next is 5 - loaded
So the probability of being shot is 1/4.
The answer
Do not spin. 1/4 beats 1/3.
Why adjacency matters
If the two bullets were placed in random chambers rather than adjacent ones, the answer flips.
Surviving one pull tells you that you are on an empty chamber, and with random placement the conditional probability that the next is loaded becomes 2/5 - worse than the 1/3 from spinning. There you should spin.
The adjacency clusters the danger, so surviving one click means you have probably moved past the loaded pair rather than toward it.
The mistake to avoid
Computing the unconditional probability that chamber 5 comes up. You must condition on having survived, which eliminates chambers 5 and 6 from your possible current position. Forgetting to condition is the standard error here.
See Bayes' theorem and conditioning, and practise in conditional probability.