Bomb Defusal

You are a bomb disposal expert. Infront of you is a bomb with an active countdown timer. You have only a few seconds to decide what to do. Hollywood movies have taught you all you need to know: there is always a red wire and a blue wire! Cutting the correct wire will stop the timer. Cutting the wrong wire will immediately set off the device. You carefully search the device and identify the red and blue wires. Which wire do you cut?

Absent any additional information, it’s a coin flip. With no other data you have a 50:50 chance of snipping the correct wire. Good Luck.

Luckily for you, you are part of an elite crime fighting unit, and have backup. Three of your colleagues have gone out and each has apprehended a henchman of the bomb designer. These henchmen all helped build the bomb and so know the correct wire to cut. Under swift interrogation, all three of them have confessed “To diffuse the bomb, you must cut the red wire”. Unfortunately, henchmen are known for not always telling the truth. Each henchman, independently, only tells the truth 75% of the time. Your colleagues only got the chance to ask this one question (nothing else to compare against), and the henchman are far enough away from the device so that they are safe whatever happens (you can’t infer they would be biased to save their own skins). Which wire should you cut now, and what are your chances?

What are your chances of survival now that you have this additional information?

Let’s draw a tree of the possibilities. There are two ways you would hear the phrase “You must cut the red wire”. The first is if it really is the red wire that needs cutting, and all three henchmen told the truth. The second case is if it is the blue wire that needs cutting, and all three henchmen lied.

Because we heard “You must cut the red wire”, we know it’s one of these two possibilities, and the chance of surviving is the ratio of the ‘correct’ cut over all possible cuts. The denominator is the sum of both probabilities, and the numerator is the probability of the correct cut. (Because we have no other information, we’re assuming the actual base chance of it being the red wire is 50%).

In this situation, the statements from the henchman have improved in our chances of the cutting the red wire and surviving to 96.4% (Go team!)

As noted above, this calculation was based on the assumption that the bomb architect selected his/her wire color choice at random. What if this were not the case? What if he had an affinity/preference for one particular wire to be the critical one? Let’s give the probability that they engineer the bomb requiring the red wire to be cut with probability p. This means that the blue wire will be used with probability (1-p).

This modifies the tree:

The new answer for the chance of surviving is 27p/(26p+1). Confirming this is correct, if we set p = 0.5 we get the answer 0.964, which is what we calculated above.

If our bomber has been naughty and made other bombs previously, then the post-mortem forensics of those bombs might be available. If he has made 12 previous bombs, and 10 of these have used the red wire, we can speculate an appropriate value for the probability of his selecting the red is p = 10/12. Plugging this into our equation above, this improves our chances to ≈ 99.26% (Go Bayes theorem!)

Below is a graph showing the relationship between the probability of success for p ranging from 0-1 (again assuming all three henchmen said “cut the red wire”).