# Where should you shoot first for the highest chance of survival?

+1 vote
116 views

Amar, Akbar and Anthony plans for gun fighting and eachone got a gun to shoot each other until only one person is left.

Shooting History of all three
Amar hits his shot 1/3 of the time, gets to shoot first.
Akbar, hits his shot 2/3 of the time, gets to shoot next if still living.
Anthony having perfect record at shooting(100% accuracy) shoots last , if alive.

If you are Amar, where should you shoot first for the highest chance of survival?

posted Jul 2, 2014

Amar should shoot Anthony first. This increases his change for survival from 50/189 to 59/189.
.
Suppose A shoots B first.
1/3 A hits B and 100% that C then kills A. (= 0% that A wins)
.
2/3 A misses B
followed by 2/3 B kills C, leaving just A and B standing.
A and B then have a shootout with A having a 3 in 7 chance of winning and B having a 4 in 7 change of winning (= 2/3 * 2/3 * 3/7 = 36/189 that A wins
.
OR after A misses B, 1/3 B misses C, followed by 100% C kills B, leaving just A and C standing.
A then has a 1/3 of killing C. If A misses, C will kill him (= 2/3 * 1/3 * 1/3 = 14/189 that A wins
.
Combining the chances that A wins gives (14 + 36 = 50/189)
.
[The shootout between A and B is 1/3 (=3/9) that A kills B immediately. If A misses, B has a 2/3 chance of killing B, so 2/3 * 2/3 = 4/9 that B wins. There is also a 2/9 chance that both miss. This series converges into 3 in 7 for A winning and 4 in 7 for B winning.
...
Alternatively, A can try to shoot C.
1/3 A kills C, followed by 2/3 that B kills A.
That leaves 1/9 that only A and B are left. A then has a 3/7 chance of winning (= 1/9 * 3/7 = 9/189 that A wins)
.
If A misses (2/3 chance), there is a 2/3 chance that B kills C, leaving only A and B. The total chance of A winning this series = 2/3 * 2/3 * 3/7 = 36/189 that A wins
.
OR B misses C (1/3 chance), leaving C to kill B. At this point only A and C remain and A has 1/3 of killing C. Total chance of A winning this series =2/3 * 1/3 * 1/3 = 14/189
.
Combining the chances that A wins gives (9 + 36 + 14 = 59/189)

Similar Puzzles

A fair coin is tossed until a head comes up for the first time. What is the probability of this happening on an odd number tosses?

At Probability University, there are 375 freshmen, 293 sophomores, 187 juniors, & 126 seniors. One student will randomly be chosen to receive an award.

What percent chance is there that it will be a junior? Round to the nearest whole percent.

A chemistry teacher took exam of his students. Out of the total students, 25% of the students passed both the tests included in the exam. However, only 42% were able to clear the first test.

Can you find out the percentage of those students who passes the first test and also passed the second test ?