What is the probability of A winning in a pistol duel against B?

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Discussion Overview

The discussion revolves around calculating the probability of player A winning a pistol duel against player B, given their respective chances of hitting the target. Participants explore different interpretations of the problem, the mathematical formulation of the solution, and alternative approaches to the probability calculation.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the problem and an initial solution involving a summation of probabilities, questioning the meaning of the terms in the expression.
  • Another participant explains the terms in the summation, detailing the probabilities of hitting and missing for both players and how they contribute to the overall probability of A winning.
  • A third participant expresses agreement with the explanation provided but does not add new information.
  • One participant challenges the interpretation of the shooting order, suggesting that if A hits, B should not be able to shoot back, indicating a misunderstanding of the problem's setup.
  • Another participant proposes an alternative method to calculate the probability using a recursive approach, defining "p" as the probability of the first player winning and formulating an equation based on the outcomes of the shots.
  • One participant assumes that both players shoot simultaneously, leading to a scenario where if both hit, there is no winner.

Areas of Agreement / Disagreement

Participants express differing interpretations of the shooting order and the implications for the probability calculation. There is no consensus on the correct understanding of the problem or the best method to solve it.

Contextual Notes

Participants have not fully resolved the assumptions regarding the shooting order and whether players shoot simultaneously or in turns, which affects the probability calculations presented.

Who May Find This Useful

Readers interested in probability theory, game theory, or mathematical modeling of competitive scenarios may find this discussion relevant.

KFC
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I read a funny question about probability in a magazine. Here is the problem. A and B are to fight pistol duel. They are to fire at their choice of target in succession in the order A and B cyclically until only man is left unhit. A's chance of hitting B is 0.3, B's is 0.5. What's the probability for A to win finally? Here is the solution

[tex]0.5*0.3 + 0.5^2*0.7*0.3 + 0.5^3*0.7^2*0.3 + \cdots[/tex]

I am trying to understand this expression. A goes first. For the first term, A have 0.3 to shot B and 0.5 of not being killed. So if A can't kill B and not being killed in the first round. Then A is going to shot B again, so there exists the second term. But how do you understand this term? Why there is only one 0.3 and two 0.5? What does the 0.7 represent?
 
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Each term in the summation is the probability of A killing B in the [tex]k[/tex]th shot. For A to kill B in the first shot, A has to hit, probability .3. B has to miss, probability .5.

For A to kill B in the second shot, A has to miss and B has to miss on the first shot. Probabilities .7 and .5. Then A has to hit and B has to miss on the second shot, probabilities .3 and .5 respectively. These events are all independent so you multiply the probabilities.

Repeat ad nauseum... each additional shot means one more miss for each of A and B, giving an extra .7 and .5

As to why it makes sense to add all these numbers up, I'll let you think about it
 
make sense, thanks
Office_Shredder said:
Each term in the summation is the probability of A killing B in the [tex]k[/tex]th shot. For A to kill B in the first shot, A has to hit, probability .3. B has to miss, probability .5.

For A to kill B in the second shot, A has to miss and B has to miss on the first shot. Probabilities .7 and .5. Then A has to hit and B has to miss on the second shot, probabilities .3 and .5 respectively. These events are all independent so you multiply the probabilities.

Repeat ad nauseum... each additional shot means one more miss for each of A and B, giving an extra .7 and .5

As to why it makes sense to add all these numbers up, I'll let you think about it
 
Hmm, that's weird. I'd rather expect that after one shoots and hits, the other is dead and can't shoot back. Your solutions seems to indicate that A is going second!

You can also solve the question another way. Let "p" be the probability for the first player to win when it's his turn. His chance to hit is p_1. If he misses, the other one gets the chance p_2 to win. So what should "p" be? It is the chance that you hit with your first shot OR
- you miss
- the other one misses
- "you win the remaining game being in the position to shoot first" (which is exactly "p" again)

So we can write down
[tex]p=p_1+(1-p_1)(1-p_2)p[/tex]
Now can solve this for p.
 
Last edited:
I assumed that they were shooting at the same time... if both die then nobody wins
 

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