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Recently I was given the following problem.

Anti-tank mines are placed on a straight line 15 meters apart from each other. The tank, 3 meters wide, runs perpendicular to this line. What is the probability that the tank will hit a mine.

My problem with this question is that the exact length of the line is not given. I guess depending on the length of the line, the probability might be different. It seems to me that the problem is not well defined. What is your opinion?

Parcly Taxel
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Sasun Grigoryan
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  • If the number of such mines is large, I think the answer should be 0.4 – ab123 Sep 15 '17 at 08:14
  • A more interesting problem is [Buffon's needle] (https://en.wikipedia.org/wiki/Buffon%27s_needle) – ab123 Sep 15 '17 at 08:16
  • The probability distribution of the point where the tank crosses is needed. Even if it assumed to be uniform, you are correct, you need the length of the line. And then whether there are mines at the ends of the line or how the mines are placed on the line. But all in all, I think the question is considering a limiting case where the line length $\to \infty$. Then the line length and exact placement of the mines don't matter any more, (assuming the uniform distribution for the line of course). – ploosu2 Sep 15 '17 at 09:40

3 Answers3

2

The following are implicitly assumed:

  • the row of mines is infinite, with mines 15 metres apart
  • the tank's position is continuously distributed

Then we can restrict the scope of the tank so its left edge runs between two adjacent mines; it never detonates the left mine. Say the tank is moving forward to an east-west row of mines; if and only if the left edge is less than 3 metres from the right mine will it detonate the right mine. The probability is thus $3/15=0.2$.

Parcly Taxel
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  • thanks for your time. But I have a question with regards to this statement - "it never detonates the left mine". Why do we rule out the possibility of the tank to hit the left mine? I mean, what is wrong with considering both the left and right mines like in satish ramanathan's answer? – Sasun Grigoryan Sep 16 '17 at 09:58
  • @SasunGrigoryan The other two answers use the centre of the tank as its position. That centre would have to be within 1.5 - not 3 as in the .4 answer - metres from a mine in order for the tank to hit the mine. – Parcly Taxel Sep 16 '17 at 10:05
  • Since the line of mines is infinite, can't the center of the tank be to the left of the left mine and still detonate? I mean in that case the total distance would be 15 plus 3 (1.5 from each side) and the distance of detonating would be 3 plus 3 (1.5 from the left ans right for each 2 bombs). By this reaaoning the probaility would be 6/18. – Sasun Grigoryan Sep 16 '17 at 10:18
  • @SasunGrigoryan The total distance is 15 metres, and the distance of detonating is 3 metres. Everything else is wrong here. – Parcly Taxel Sep 16 '17 at 11:44
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The distance of 3 meters to each side of a mine would equal to 6 for each mine. The total distance is 15 between each mine. The tank would be blown away within this 6 meters distance around each mine and hence the probability is $\frac{6}{15} = .4$

Satish Ramanathan
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  • Take from midpoint of two mines to another mide point of adjacent two mines. The distance is 15 meters. Within these 15 meters, the mine common to both would sit in the middle. Take 3 to the left and 3 to the right. that will total it up to 6 and the distance from mid to mid is 15 as it would be from mine to mine. – Satish Ramanathan Sep 15 '17 at 08:57
  • Seems to be a legit solution for just 2 mines but what if he there are more than 2 mines? – Sasun Grigoryan Sep 15 '17 at 09:00
  • Even for 2 mines depending on the length of the line the probability changes, right? – Sasun Grigoryan Sep 15 '17 at 09:02
  • Draw a line with mines every 15 meters and draw around each mine 3 to the left and 3 to the right. Do a series of mines and you will find that it does not matter if it is just two mines or 10 mines, the tank goes perpendicular to this line and has the same probability whenever it crosses the line with mines – Satish Ramanathan Sep 15 '17 at 09:05
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The true tank doesn't crawl, dragging its bottom over the ground; it rather rides on wheels, which roll over two caterpillar tracks. So it's actually ill-posed (or rather falsely modelled) situation, you would need to know the track width together with the distance between tracks, not the whole vehicle's width.

However, assuming the simplified problem, you may consider the position of the tank's centre over the mine line. If the centre falls inside a 3-meter segment centered over any mine, the tank gets destroyed, otherwise it crosses the line untouched. There is one 3-meter death zone per every 15-meter segment between mines, so the probability of hitting a mine is $3/15 = 1/5$.

CiaPan
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