# WWIII

Pakistan, India and Shir Lanka are engaged in a missile war, such that they fire one after the other, in a sequence, the sequence selected by draw of lots. The rules of war are such that the 3 countries keep on shooting at each other until they hit with the missile. This goes on until there is one last country left unhit, and then the war ends. The chances that Pakistan hits its target is 100%, Shir Lanka hits the target is 80% and India hits the target is 50%.

Who has the highest chance of winning the war?

### 2 guesses to WWIII

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• brent

Well, if my logic, math & understanding of the riddle are all correct, I get:
India has the greatest chance (44.7%)
Sri Lanka next @ 31.1%
and Pakistan has 24.2%

One assumption I made is that the country firing will always target the most accurate other country.

Interestingly, India could further improve its chances by intentionally missing all targets when both other countries are still in. This is because I assume india will target pakistan, but if they are successful, Sri Lanka then has an 80% chance of winning over India on the next shot (and an 88.9% chance of winning against india over all)

Where if india were to just waste the shot, Either Pakistan would eliminate Sri Lanka – leaving India with a 50% chance shot at winning the war, OR Sri Lanka would have an 80% chance of eliminating Pakistan, which leaves india with a 55.6% chance of winning against Sri Lanka. The 20% of the time Sir Lanka misses pakistan, Pakistan follows up by hitting Sir Lanka, and again, India then has a 50% chance against Pakistan.

• Brent, you are today’s winner!
India has the highest chances of winning the war (47/90) Missile Firing order: Chances of Remaining Unhit – Pakistan: Chances of Remaining Unhit – Srilanka: Chances of Remaining Unhit – India. Pakistan – Srilanka – India 1/2 0 1/2. Pakistan – India – Srilanka 1/2 0 1/2. India – Pakistan – Srilanka 1/10 16/45 49/90. Srilanka- Pakistan – India 1/2 0 1/2. Srilanka – India – Pakistan 1/10 16/45 49/90. India – Srilanka – Pakistan 1/10 16/45 49/90. Total chances of being unhit (sum of the probabilities divided by 6): 27/90 16/90 47/90