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## Question

bonanova    76

Well now, don't that just give the place a new look!

Alex had just hung a new dartboard on the wall and

was stepping back to admire his handiwork. Ian and

Jamie actually weren't too sure, and Davey hardly

looked up from where he sat at the bar.

Well it's kind of diff'rent, now, isn't it? allowed Jamie.

Yeh, it looks simple - no doubles or triples or anything

agreed Ian. And just multiples of 5, added Davey, who

became curious enough to walk over. There's a lot of

scores no one could make on that board ... so, what's

the point?

That's exactly the point, beamed Alex. Simplicity!

Usually you blokes don't like my propositions, cuz they're

a little bit beyond yer brains don't ya know. So I'm making

this one really easy to understand. Ian started twitching,

and Jamie shuffled his feet a bit. They'd heard pitches

from Alex before, and hearing that this one was simple

I'm thinking of a number, continued Alex, an easy one: 25.

You get two darts and all you have to do is score 25 points.

Look at the combinations that win for you. Even odds,

and I'll even take away the out of bounds. If you don't

hit a number, you throw another dart. Two darts that

score, and the score has to be 25. Who's ready for a

challenge?

When the boys were silent, Alex stepped up the pitch.

All right, two to one odds your quid against my two. Like

taking candy from a baby, but you caught me in a good

mood tonight. Ian shook his head, Jamie sat down to

think it over, and Davey sauntered back to the bar.

After a moment, Davey hollered, make it 3 to 1 and

you're on. Ouch! cried Alex, if I weren't such a sport

I might not agree, but hey - we're friends, aren't we?

Let's see what ya got!

Assuming that Davey was no great shakes at aiming

the darts, with the only thing assured being that two

darts end up scoring points, did Davey make a good

bet? Or did Alex do his friends in, one more time?

The radii of the four circles are as shown: 1, 2, 3 and 4

in some units.

## 5 answers to this question

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Guest

Percentages for each circle as follows:

5pt: 43.75%

10pt: 31.25%

15pt: 18.75%

20pt: 6.25%

Only combinations that are good are:

5 then 20 - 2.7344%

10 then 15 - 5.89594%

15 then 10 - 5.8594%

20 then 5 - 2.7344%

Yielding a percent of 17.1875% assuming the guy is just throwing randomly.

He should expect to complete the task in 5.82 tries. Thus Alex has the advantage again

Edited by PolishNorbi

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Guest

Davey should say the following,

"After careful consideration, I want to remove the offer of 3 to 1 and propose a counter-offer of give me an extra dart and keep the same rules. Only two darts count (any two that equal 25 is a winning score). And misses don't count.

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Guest

Alex totally took his friends in again. There are 10 possible outcomes. Weighting those by the total combined area for each combination of darts (you can remove pi and just use r^2 since it keeps everything proportional) yields you a ratio of 4:1 against totaling 25.

A20 = 1

A15 = 2^2 - A20

A10 = 3^2 - A15

A5 = 4^2 - A10

Totaling 25:

A5 + A20 = 8

A10 + A15 = 9

Total = 17

Missing:

A5 + A5 = 14

A5 + A10 = 12

A5 + A15 = 11

A10 + A10 = 10

A10 + A20 = 6

A15 + A15 = 8

A15 + A20 = 5

A20 + A20 = 2

Total = 68

Of course, Alex is seriously depending on random aim. I bet Dave's odds would increase significantly if he just aimed for the 20, since he's not likely to hit it, he'll probably end up landing them both in the 10's and 15's.

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Alex totally took his friends in again. There are 10 possible outcomes. Weighting those by the total combined area for each combination of darts (you can remove pi and just use r^2 since it keeps everything proportional) yields you a ratio of 4:1 against totaling 25.

A20 = 1

A15 = 2^2 - A20

A10 = 3^2 - A15

A5 = 4^2 - A10

Totaling 25:

A5 + A20 = 8

A10 + A15 = 9

Total = 17

Missing:

A5 + A5 = 14

A5 + A10 = 12

A5 + A15 = 11

A10 + A10 = 10

A10 + A20 = 6

A15 + A15 = 8

A15 + A20 = 5

A20 + A20 = 2

Total = 68

Of course, Alex is seriously depending on random aim. I bet Dave's odds would increase significantly if he just aimed for the 20, since he's not likely to hit it, he'll probably end up landing them both in the 10's and 15's.

Nope. My solution was flawed. PolishNorbi's approach was simpler. I was trying to avoid problems that could result from failing to consider the possibilities together, since order is unimportant. That's why I suggested only 10 possible outcomes, while there are 16 if you include order. Oh well.

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I knew alex could do it

+ u guys are good at math

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