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Guest Message by DevFuse

# gun with unlimited bullets

Best Answer bonanova, 08 May 2014 - 10:19 AM

Spoiler for Final thoughts

Go to the full post

12 replies to this topic

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Posted 19 March 2014 - 03:48 AM

There's a gun located on an infinite line. It starts shooting bullets along that line at the rate of one bullet per second. Each bullet has a velocity between 0 and 1 m/s randomly chosen from a uniform distribution. If two bullets collide, they explode and disappear. What is the probability that at least one of the bullets will infinitely fly without colliding with another bullet?
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### #2 harey

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Posted 19 March 2014 - 10:06 AM

Spoiler for

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### #3 santhu221633

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Posted 19 March 2014 - 11:49 AM

Spoiler for

what you said is wrong. For your understanding, what if first bullet has velocity 1m/s, then it will reach infinity right.

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### #4 Pickett

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Posted 19 March 2014 - 01:27 PM

Spoiler for Well

Edited by Pickett, 19 March 2014 - 01:35 PM.

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### #5 Pickett

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Posted 19 March 2014 - 01:39 PM

Spoiler for A bit more explanation for my above

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### #6 bonanova

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Posted 19 March 2014 - 05:00 PM

Spoiler for Depends

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Vidi vici veni.

### #7 Rainman

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Posted 20 March 2014 - 04:39 PM

Spoiler for I remain unconvinced

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Posted 20 March 2014 - 09:07 PM

i changed the status of this question as we consider Rainman's argument.

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### #9 bonanova

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Posted 21 March 2014 - 05:05 AM

Spoiler for I remain unconvinced

Your conjecture of 0.02 probability of reaching parity of heads and tails for a coin with p(H) = .99 is correct.

A biased coin with probability pmin of the less-likely outcome behaves, with probability 2pmin, like an unbiased coin.

I'm not sure which case you're unconvinced of, (0, 1) or [0. 1].

Since the closed case seems clear, here's an expanded description of  the open case:

Spoiler for For open interval

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Vidi vici veni.

### #10 bonanova

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Posted 11 April 2014 - 11:38 AM

Revised answer, without need of spoiler:

I now believe (with Rainman,) but cannot prove it, that in general bullets will escape.

1. If first bullet has v=1 (possible but with zero probability) it will escape.

2. Since bullets destroy themselves pairwise, If we could show that infinity were "even" then an even number of bullets will escape (including possibly zero.) If we could show that infinity were "odd" then an odd number of bullets will escape (guaranteeing at least one.) But infinity is not a number and does not have parity. There may be some validity to averaging the results to say that (2n-1)/2 bullets are guaranteed to escape, but I think that is specious. It's like saying the alternating series 1 - 1 + 1 - 1 + 1 - 1 + ... converges to 1/2. The partial sums are alternately 1 and 0, and in fact it does not converge at all.

3. The question that I do not know how to answer is how efficiently a faster bullet can destroy the leading bullet if there are intervening bullets. It seems too complicated, even, to simulate. If the leading bullet has speed very close to 1, the expected number of bullets before a faster one is fired will be very large. In my previous post I said they would self-annihilate with non-zero probability. But that's not enough. If no bullet is to escape, this must happen with certainty. It is not likely that the intervening bullets will [a] be even in number and [b] destroy each other, which would be necessary for the leading bullet to be caught. This leads me to agree with Rainman.

It seems that with a high degree of likelihood bullets will escape. But it is not a proof.

This is a good puzzle.

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Vidi vici veni.

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