Probability that randomly-thrown dart lands inside....

[Monkeyeatbutt]

In the figure below, each of the 4 circles has a radius of 2.5 feet. If each circle just touches 2 other circles as well as 2 sides of the square, what is the probability (to the nearest whole percent) that a randomly-thrown dart lands inside the square and will land in one of the shaded circles?

The picture is just a white square with 4 circles touching, kinda like this:

iooi
iooi

But it's bigger, and everything is touching.

I know the math needed to work it out -- I have an example sitting in frount of me. But the example is for a target (they take out the white rings) and they know how big the box is. I'm not sure how to start with out the other info.

 

[soroban]

Re: Probability

Hello, Monkeyeatbutt!

Thanks for your diagram . . . it really helped.

And I bet you do know how to answer this question . . .

In the figure below, Each of the 4 circles has a radius of 2.5 feet.
If each circle touches 2 other circles as well as 2 sides of the square.
what is the probability (to the nearest whole percent) that a randomly thrown dart
lands inside the square and will land in one of the shaded circles?

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The radius of each circle is 2.5 feet, so its diameter is 5 feet.

Look at the diagram . . . the square is 10 feet on a side.
$\displaystyle \;\;$Its area is: $\displaystyle \,10^2\,=\,100$ ft².

Each circle has a radius of 2.5 feet.
$\displaystyle \;\;$Its area is: $\displaystyle \,\pi r^2\;=\;\pi(2.5^2) \:=\:6.25\pi$ ft².
Then four circles have an area of: $\displaystyle \,4\,\times\, 6.25\pi \,=\,25\pi$ ft².

And the probability is the ratio of the areas, right?

$\displaystyle \;\;\;$So the answer is: $\displaystyle \;\frac{25\pi}{100}\:\approx\:79\%$


See? . . . You knew all that, didn't you?

 

[Monkeyeatbutt]

thank you so much I think I get it now.