Question

Is she a skilled player for hitting the bull’s eye or is the result of hers expected?

Answer 1

You play darts with a friend. The board is a circle with a 6-inch radius. The “Bull’s Eye” is a smaller circle in the center of the board with an 0.25-inch radius. You throw darts (small metal arrows) at the board.

At the end of six weeks, she has thrown 500 darts. She hit the board every time and hit the Bull’s Eye 1 time

If you throw 500 darts, how many times should you expect to hit the Bull’s Eye by random chance?

Compare this result to your friend’s result.

Is she a skilled player for hitting the bull’s eye or is the result of hers expected?

Answer 2

What kind of distribution are your lessons about? this is a pretty easy problem in a uniform distribution (likely to hit anywhere on the board), tough in a "normal" distribution (most likely to hit one spot...)

Answer 3

Well since the bullseye is considerably smaller than the rest of the board I'd assume this isn't a uniform problem. I'm not sure if that's what you're asking though

Answer 4

If it's geometry, it means you need to find the area of the bullseye, then find the area of the dartboard, then divide them.

Answer 5

oh okay! thank you

Answer 6

I get 5.76, so what do I do with it? Is that the probability of hitting the bull's eye on average?

Answer 7

I don't know what that is; show your work.

Answer 8

Same as @agent0smith , we should look at uniform and figure what the relationship is between the the bullseye and the whole board.
But I'll make a quick sidebar:
Example of uniform probability: you put a plate out in the rain. a raindrop is just as likely to hit anywhere on the plate.
Example of normal distribution: bell curve, like if you shoot a gun at a target. most shots will hit near the center and a few, less likely shots will hit farther out.

Answer 9

Area of the board is 113.04 (6^2 * 3.14) and area of the bull's eye is 19.625 (0.25^2 * 3.14), 19.625/113.04 = 5.76

Answer 10

Ah, yeah, then it's uniform probability.

Answer 11

19.625/113.04 = 5.76

Does that seem reasonable?

Answer 12

err sorry 113.04/19.625

Answer 13

Probability is never greater than 1. Should tell you that you're doing something wrong.

Answer 14

I figured I was dividing the wrong way around

19.625/113.04 = 0.174

Answer 15

That's the probability.

Now
If you throw 500 darts, how many times should you expect to hit the Bull’s Eye by random chance?

Answer 16

0.87 times?

Answer 17

You're throwing 500 darts.

On each throw, your chance of bullseye is 0.174, or 17.4%

Does 0.87 times seem reasonable?

Answer 18

oh I see

Answer 19

So 87 times out of 500?

Answer 20

Show your work so I know if you're doing it right.

Answer 21

Wait, 17.4% of 100 is 17.4, so is 17.4 * 5 what I'm looking for?

Answer 22

That works :)

Answer 23

87/500 chance then

Answer 24

Remember the question asks
If you throw 500 darts, how many times should you expect to hit the Bull’s Eye by random chance?

Answer 25

So how does it compare to your friend?

To put it nicely, she sucks a.ss.

Answer 26

Hahahaha yep! Thank you very much for your help.

Answer 27

Welcome.

Answer 28

@junebuggie12 0.25^2 * 3.14 that cannot be 19...

Answer 29

You left too quickly but that being wrong has made your final answer wrong.

Answer 30

Oh, it's 0.19625... not 19.625.

Answer 31

Important principle: probabilities are never less than zero nor greater than one:
\[0\le x \le 1\]

Answer 32

So you should know that a probability result such as 19.625 must be discarded immediately. On the other hand, 0.19625 could be a probability.

Answer 33

@mathmale it wasn't her probability that was off, it was just the area of the bullseye.