Question

Let $A$ be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0,60]$ is less than or equal to a If $P(A)=\frac{11}{36}$, then a is equal to _____

Answer 1

Correct Answer: 27

Let \( A \) be the event that the absolute difference between two randomly chosen real numbers in the sample space \( [0, 60] \) is less than or equal to \( a \). If \( P(A) = \frac{11}{36} \), then \( a \) is equal to ______.

Step 1: Define the problem
We are given that the two numbers are randomly chosen from the interval \( [0, 60] \), and we need to find the value of \( a \) such that the absolute difference between these two numbers is less than or equal to \( a \).

Step 2: Probability setup
Consider two real numbers, \( x \) and \( y \), chosen from \( [0, 60] \). We want to find the probability that \( |x - y| \leq a \), where \( x \) and \( y \) are independent. The total area of the sample space is the area of a square with side length 60, i.e., \( 60 \times 60 = 3600 \).

The correct answer is 50.
∣x−y∣<a⇒−a<x−y<a 
⇒x−y<a and x−y>−a 
 

P(A)=(OBDF)ar(OACDEG)​ 
=ar(OBDF)ar(OBDF)−ar(ABC)−ar(EFG)​ 
⇒3611​=3600(60)2−21​(60−a)2−21​(60−a)2​ 
⇒1100=3600−(60−a)2 
⇒(60−a)2=2500⇒60−a=50 
⇒a=10