Investigate the difficulty of achieving an outshot of 141 in a game of 501

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J_K1

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Hi,

I need some help on my maths homework. It is about darts. I need to investigate the difficulty of achieving an outshot of 141 in a game of 501.

- Display the sample space
- Describe the events (possible outshots of 141)
- Calculate the probability of achieving each of the possible out shots with a score of 141
 
Harry_the_cat said:
What have you done so far?
Click to expand...
I have worked out some outcomes to achieve 141

Triple/Triple/Double:
19 (57), 20 (60), 12 (24) = 141
17 (51), 20 (60), 15 (30) = 141
15 (45), 20 (60), 18 (36) = 141
19 (57), 18 (54), 15 (30) = 141
19 (57), 16 (48), 18 (36) = 141
18 (54), 17 (51), 18 (36) = 141

Double/Triple/Bullseye
20 (40), 17 (51), 1 (50) = 141
17 (34), 19 (57), 1 (50) = 141
 
I doubt the problem is meant to do by hand. It requires a computer program.
  • Anyhow, here's the sample space from Wikipedia.
Screen Shot 2022-02-23 at 7.32.13 AM.png
  • List for all combinations that add up to 141 with 3 throws.
  • The probability of each outcome is calculated as the area of sector/area of the dartboard based on the actual dimensions of the dartboard.
  • Next, assuming each throw is independent, the probability of each combination is found by multiplying the probability of each outcome of that combination.
    • Using one combination you listed as an example, 57+60+24=141. The probability of this combination is [math]\Pr(57 \cap 60 \cap 24)=\Pr(57)\Pr(60)\Pr(24)[/math]
  • Lastly, summing up all the probability of all combinations will give you the final answer.
 
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@J_K1
As I stated before, this is more like a project which requires you to answer multiple questions, as I laid out for you in post #5, rather than just a question. The manual calculation is quite tedious and repetitive, but it's not impossible.
  • Are you allowed to use a computer program/coding?
  • To calculate the area of the bullseye (50 points) and double bullseye (25 points), you need the radius of the 2 circles and then apply the formula for the area of a circle.
  • How are you planning on calculating the rest of the areas?
 
BigBeachBanana said:
@J_K1
As I stated before, this is more like a project which requires you to answer multiple questions, as I laid out for you in post #5, rather than just a question. The manual calculation is quite tedious and repetitive, but it's not impossible.
  • Are you allowed to use a computer program/coding?
  • To calculate the area of the bullseye (50 points) and double bullseye (25 points), you need the radius of the 2 circles and then apply the formula for the area of a circle.
  • How are you planning on calculating the rest of the areas?
Click to expand...
Yes i am allowed to use a computer program/coding but i dont know how to
 
BigBeachBanana said:
@J_K1
As I stated before, this is more like a project which requires you to answer multiple questions, as I laid out for you in post #5, rather than just a question. The manual calculation is quite tedious and repetitive, but it's not impossible.
  • Are you allowed to use a computer program/coding?
  • To calculate the area of the bullseye (50 points) and double bullseye (25 points), you need the radius of the 2 circles and then apply the formula for the area of a circle.
  • How are you planning on calculating the rest of the areas?
Click to expand...
I have calculated the rest of the areas, is just that i got confused which formula do i use for the area of the bullseye
 
J_K1 said:
I have calculated the rest of the areas, is just that i got confused which formula do i use for the area of the bullseye
Click to expand...
I would think the area of the bullseye is easier to calculate than any of the other areas. Use the formula for the area of a circle. How did you calculate the other areas?
 
BigBeachBanana said:
I doubt the problem is meant to do by hand. It requires a computer program.
  • Anyhow, here's the sample space from Wikipedia.
View attachment 31270
  • List for all combinations that add up to 141 with 3 throws.
  • The probability of each outcome is calculated as the area of sector/area of the dartboard based on the actual dimensions of the dartboard.
  • Next, assuming each throw is independent, the probability of each combination is found by multiplying the probability of each outcome of that combination.
    • Using one combination you listed as an example, 57+60+24=141. The probability of this combination is [math]\Pr(57 \cap 60 \cap 24)=\Pr(57)\Pr(60)\Pr(24)[/math]
  • Lastly, summing up all the probability of all combinations will give you the final answer.
Click to expand...
That's the sample space for a single shot. Wouldn't the required sample space be all the possiible totals for 3 shots?
 
Harry_the_cat said:
That's the sample space for a single shot. Wouldn't the required sample space be all the possiible totals for 3 shots?
Click to expand...
Yes, you're correct. The diagram only shows the sample space of 1 shot. To list the sample space for 3 shots, the OP requires to identify the exhaustive combinations for 3 shots.
J_K1 said:
Yes i am allowed to use a computer program/coding but i dont know how to
Click to expand...
@J_K1 Are you expected to do this by hand? I'm just confused as to why your instructor would assign such a project if you don't know how to code. What class is this from?
 
BigBeachBanana said:
Yes, you're correct. The diagram only shows the sample space of 1 shot. To list the sample space for 3 shots, the OP requires to identify the exhaustive combinations for 3 shots.

@J_K1 Are you expected to do this by hand? I'm just confused as to why your instructor would assign such a project if you don't know how to code. What class is this from?
Click to expand...
Yes, i was expected to do it by hand
 
J_K1 said:
Yes, i was expected to do it by hand
Click to expand...
IMO, I think it's unfair for you to do it by hand so I'll give you the list of the combinations. This is also your sample space. There are 23 of them:

(21,60,60);(24,57,60);(27,54,60);(27,57,57);(30,51,60)
(30,54,57);(33,48,60);(33,51,57);(33,54,54);(34,50,57)
(36,45,60);(36,48,57);(36,51,54);(39,42,60);(39,45,57)
(39,48,54);(39,51,51);(40,50,51);(42,42,57);(42,45,54)
(42,48,51);(45,45,51);(45,48,48)
I hope you're keeping a spreadsheet to save your work.
 
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Sorry, for bothering, thank you for your suggestions given but the teacher requires us to have 20 possible out shots of 141, its like T DT, it cannot be like TTT . And the combination you have provided is the same one that I gave you from the beginning. It may seems to be a little confusing. Anyways, thanks for helping me out.
 
J_K1 said:
Sorry, for bothering, thank you for your suggestions given but the teacher requires us to have 20 possible out shots of 141, its like T DT, it cannot be like TTT . And the combination you have provided is the same one that I gave you from the beginning. It may seems to be a little confusing. Anyways, thanks for helping me out.
Click to expand...
I gave you all the possibilities, of course, it's gonna include the one you provided. If you don't want TTT, then just remove them?
 
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