MHB Ken's Question from Yahoo Answers: Probability Question For Verification?

  • Thread starter Thread starter CaptainBlack
  • Start date Start date
  • Tags Tags
    Probability
CaptainBlack
Messages
801
Reaction score
0
Question:
"A point \({\rm{P}}(x,y)\) is chosen at random in a unit disc, centred at \((0,0)\).

The probability required is that the point chosen is such that both \(| x -y| \lt 1\) and \(|x+y| \lt 1\) .

Is the answer \(2/\pi\) or \(1-2/\pi\)?

Thank you."Answer:
I take disc to be a disc of radius 1 centred at the origin.

The region defined by the inequalities \(|x-y| \lt 1\) and \(|x+y| \lt 1\) is an inscribed square to the circle, which has side \(\sqrt{2}\) and hence area \(2\). The area of the circle is \(\pi\), so the probability that a point sampled uniformly on the unit disc satisfies the inequalities is the ratio of these two area: \(2/\pi\).

To convince yourself that the required region is the interior of the square rather than the exterior consider the point \((0,0)\), does it satisfy the inequalities. It it does then you want the interior of the square rather than the exterior.

Below is a scatter plot showing random points uniformly sampled on the unit disc and in black those satisfying the inequalities:

https://lh3.googleusercontent.com/AsrqIRhjcPwGKPW6RSzDwZRoH0ryjndkugx09Ohv2VkvdbS60GwQ4Gtv2A4qZZSiWoBqxPZVPw
 
Last edited:
Mathematics news on Phys.org
4LhKv1R4M3w1jH8l0t3A6LJW1z2kQlW4oLJfKQe2-3vW4uw3RrZ-DeQ2k6n1U

So, the answer is indeed \(2/\pi\). I hope this helps!
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
1
Views
3K
Replies
1
Views
2K
Replies
8
Views
2K
Replies
1
Views
11K
Replies
5
Views
2K
Replies
4
Views
1K
Back
Top