1. ## Geometric probability.

Half of a circle is inside a square and half is outside, as shown. If a point is selected at random inside the square, find the probability that the point is not inside the circle. (r is the radius)

TYA

~Vince

2. Originally Posted by vince5002001
Half of a circle is inside a square and half is outside, as shown. If a point is selected at random inside the square, find the probability that the point is not inside the circle. (r is the radius)

TYA

~Vince
You know that:

The area of the square = $\displaystyle (2r)^2 \longleftrightarrow 4r^2$
You know that the area of the circle = $\displaystyle \pi r^2$
So the part of the circle that is IN the square = $\displaystyle \frac {\pi r^2}{2}$
So $\displaystyle (4r^2$ - $\displaystyle \frac {\pi r^2}{2})$ gives you the area of the part of the square that isn't part of the half-circle.

Can you do the rest?

3. - = $\displaystyle (4r^2)2 = 8r^2 -\pi r^2$

Then would I divide by $\displaystyle r^2$? and get $\displaystyle 8-\pi$ then substitute $\displaystyle 8$ for the area of the square?

Giving you $\displaystyle {8-\pi}/{8}$ for the probability that the point is not inside the circle but inside the square.

4. Originally Posted by vince5002001
- = $\displaystyle (4r^2)2 = 8r^2 -\pi r^2$

Then would I divide by $\displaystyle r^2$? and get $\displaystyle 8-\pi$ then substitute $\displaystyle 8$ for the area of the square?

Giving you $\displaystyle {8-\pi}/{8}$ for the probability that the point is not inside the circle but inside the square.
Factorizing $\displaystyle (4r^2) - \frac{\pi r^2}{2}$ gives me the following:

$\displaystyle \frac {2(4r^2)}{2} - \frac {\pi r^2}{2} \longleftrightarrow \frac {2(4r^2)-\pi r^2}{2} \longleftrightarrow \frac {8r^2-\pi r^2}{2}$.

We know that the area of the half-circle divided by the area of the whole square multiplied by 100 gives us the percentage that the half-circles occupies in the square. Then, 100% minus the percentage found gives us the percentage of the rest of the square, which is the probability we're looking for. But I've tried to find a concrete percentage but I couldn't, I'm a bit rusty tonight and it's getting late Tell me if you get anything new...

Those are the calculations for what I've stated above:

$\displaystyle \frac {2\pi r^2}{8r^2 - \pi r^2}$ x 100

$\displaystyle \frac {2\pi r^2}{r^2(8-\pi)}$ x 100

$\displaystyle \frac {2\pi}{(8-\pi)}$ x 100

$\displaystyle \frac {200\pi}{(8-\pi)}$% = The percentage that the half-square occupies

100% - $\displaystyle \frac {200\pi}{(8-\pi)}$%= The probability we're looking for...

$\displaystyle \frac {200\pi}{(8-\pi)} \approx$ 129%

100% - 129% = -29%

5. Originally Posted by Sodapop
Factorizing $\displaystyle (4r^2) - \frac{\pi r^2}{2}$ gives me the following:

$\displaystyle \frac {2(4r^2)}{2} - \frac {\pi r^2}{2} \longleftrightarrow \frac {2(4r^2)-\pi r^2}{2} \longleftrightarrow \frac {8r^2-\pi r^2}{2}$.

We know that the area of the half-circle divided by the area of the whole square multiplied by 100 gives us the percentage that the half-circles occupies in the square. Then, 100% minus the percentage found gives us the percentage of the rest of the square, which is the probability we're looking for. But I've tried to find a concrete percentage but I couldn't, I'm a bit rusty tonight and it's getting late Tell me if you get anything new...

Those are the calculations for what I've stated above:

$\displaystyle \frac {2\pi r^2}{8r^2 - \pi r^2}$ x 100

$\displaystyle \frac {2\pi r^2}{r^2(8-\pi)}$ x 100

$\displaystyle \frac {2\pi}{(8-\pi)}$ x 100

$\displaystyle \frac {200\pi}{(8-\pi)}$% = The percentage that the half-square occupies

100% - $\displaystyle \frac {200\pi}{(8-\pi)}$%= The probability we're looking for...

$\displaystyle \frac {200\pi}{(8-\pi)} \approx$ 129%

100% - 129% = -29%
This calculation is wrong. A probability cannot be negative.

The correct answer is $\displaystyle \frac{4r^2 - \frac{1}{2} \pi r^2}{4r^2}$ which simplifies to $\displaystyle 1 - \frac{\pi}{8}$.

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### probability of a point inside a square

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