# Probability duel

Three men - conveniently named A, B, C - are fighting a duel with pistols. It's A's turn to shoot.

The rules of this duel are rather peculiar: the duelists do not all shoot simultaneously, but take turns. A fires at B, B fires at C, and C fires at A; the cycle repeats until there is a single survivor. If you hit your target, you'll fire the next person on your next turn.

For example, A might shoot and hit B. With B out of the picture, it would be C's turn to shoot - suppose he misses. Now it's A's turn again, and he fires at C; if he hits, the duel is over, with A the sole survivor.

To bring in a little probability, suppose A and C hit their target with probability 0.5, but that B is better shot, and hits with probability 0.75 - all shots are independent.

What's the probability that A wins the duel?

Probability puzzle/Getting Serious/Puzzle 11

P.S.: Sorry if there are typos! I know the answer. This is a warm-up for the next task. I won’t tell anyone if your answer is right or wrong.

Note by A Former Brilliant Member
11 months, 3 weeks ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

I have seen this question...

I think it's $100\%$

P.S Aryan Sanghi thinks I am a $\LaTeX$ warden

- 11 months, 3 weeks ago

Where did he post that??? @Yajat Shamji