bonanova Posted February 13, 2008 Report Share Posted February 13, 2008 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 made them a bit edgy. 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 13, 2008 Report Share Posted February 13, 2008 (edited) 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 February 13, 2008 by PolishNorbi Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 13, 2008 Report Share Posted February 13, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 13, 2008 Report Share Posted February 13, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 13, 2008 Report Share Posted February 13, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 TwoaDay Posted February 14, 2008 Report Share Posted February 14, 2008 I knew alex could do it + u guys are good at math Quote Link to comment Share on other sites More sharing options...
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bonanova
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
made them a bit edgy.
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.
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