Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Mock AIME 2 2010 Problems/Problem 3"

m (→‎Solution)
m (→‎Solution 1(PIE))
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 
Five gunmen are shooting each other. At the same moment, each randomly chooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form <math>\frac{a}{b}</math>, where <math>a, b</math> are relatively prime positive integers. Find <math>a+b</math>.
 
Five gunmen are shooting each other. At the same moment, each randomly chooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form <math>\frac{a}{b}</math>, where <math>a, b</math> are relatively prime positive integers. Find <math>a+b</math>.
−
==Solution 1(PIE)==
 
−
Let the five people be <math>A, B, C, D,</math> and <math>E</math>. Let <math>A_1</math> denote the event of <math>A</math> and <math>B</math> shooting each other and <math>A_2</math> through <math>A_10</math> similarly. Then, <math>P(\text{Some 2 people are shooting each other})=P(A_1 \cup A_2 \cup \dots \cup A_10)</math>.
 

Latest revision as of 14:18, 31 December 2023

Problem

Five gunmen are shooting each other. At the same moment, each randomly chooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Find $a+b$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_2_2010_Problems/Problem_3&oldid=209216"