Probability of a triangle

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• Jun 23rd 2006, 09:30 PM
malaygoel
Quote:

Originally Posted by JakeD

\displaystyle \begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}

I tried this but could not get much far.
$\displaystyle x=cosA (0<A<\frac{\pi}{2})$
$\displaystyle y=cosB (0<B<\frac{\pi}{2})$
$\displaystyle cosA + cosB \leq 1 \leq cos^2A + cos^2B$

KeepSmiling
Malay
• Jun 24th 2006, 02:05 AM
JakeD
Quote:

Originally Posted by JakeD
Use integral calculus to find the area of the region

\displaystyle \begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}

Call this region $\displaystyle D$ and rewrite it as

\displaystyle \begin{aligned} 0 \le x &\le 1 \\ 1-x \le y &\le \sqrt{1-x^2}. \\ \end{aligned}

The probability, which is a double integral, can then be evaluated as an iterated integral

\displaystyle \begin{aligned}P &= \underset{D\ \ }{\iint} 1\ dxdy \\ &= \int_0^1 dx\ \int_{1-x}^{\sqrt{1-x^2}} 1\ dy \\ &= \int_0^1 \sqrt{1-x^2} - (1-x)\ dx \\ &= \sin^{-1}(1)/2 - (1 - 1/2) \\ &= \frac{\pi -2}{4}.

I used The Integrator to find the antiderivative of $\displaystyle \sqrt{1-x^2} .$
• Jun 24th 2006, 09:52 PM
malaygoel
Quote:

Originally Posted by JakeD
Call this region $\displaystyle D$ and rewrite it as

\displaystyle \begin{aligned} 0 \le x &\le 1 \\ 1-x \le y &\le \sqrt{1-x^2}. \\ \end{aligned}

The probability, which is a double integral, can then be evaluated as an iterated integral

\displaystyle \begin{aligned}P &= \underset{D\ \ }{\iint} 1\ dxdy \\ &= \int_0^1 dx\ \int_{1-x}^{\sqrt{1-x^2}} 1\ dy \\ &= \int_0^1 \sqrt{1-x^2} - (1-x)\ dx \\ &= \sin^{-1}(1)/2 - (1 - 1/2) \\ &= \frac{\pi -2}{4}.

I used The Integrator to find the antiderivative of $\displaystyle \sqrt{1-x^2} .$

Great Work JakeD!

I want to tell you that this problem is assigned to a student with the instructions to avoid calculus and coordinate geometry for this question(I am sorry, I should have told you this before).
You can use trigonometry and combanotrics,algebra and number theory.

Keep Smiling
Malay
• Jun 25th 2006, 02:39 AM
JakeD
Quote:

Originally Posted by malaygoel
Great Work JakeD!

I want to tell you that this problem is assigned to a student with the instructions to avoid calculus and coordinate geometry for this question(I am sorry, I should have told you this before).
You can use trigonometry and combanotrics,algebra and number theory.

Keep Smiling
Malay

What is the point of this problem? To take something very simple and make it complicated with combinatorics and number theory? Or is it to keep it simple? I think it is the latter. So I'd just go with the picture as your rough sketch and be done with it.
• Jun 25th 2006, 09:13 PM
malaygoel
Quote:

Originally Posted by JakeD
What is the point of this problem? To take something very simple and make it complicated with combinatorics and number theory? Or is it to keep it simple? I think it is the latter. So I'd just go with the picture as your rough sketch and be done with it.

Ok
Nice