Re: Grace's Game

On Nov 2, 10:38 am, Daniel Webb <daniel.cobw...@xxxxxxxxxxxxxx> wrote:
David, thanks for the reply.

Re-reading my post I understand your confusion, may I start again ?

------------------------------------------------
My daughter plays a dice game where she has to throw exactly the
right number to win. Anybody have a
minute to explain how to work out the chances of any player winning ?

Game Instructions
1. Draw a grid (eg: 4x4 for 16 squares)
2. Throw a dice, and fill in the corresponding number of squares.
3. Repeat step 2 until:
a) You throw exactly the number of empty squares - YOU WIN
b) You throw more than the number of empty squares - YOU LOSE

Eg: You have three squares left then either:
throw a three - YOU WIN!
throw less than a three - go to step 2
throw more than a three - YOU LOSE!
------------------------------------------------

One die or two dice?

one die - sorry, I never knew dice was plural

Are you worried that the first person to throw may have some advantage or
disadvantage, because otherwise it looks pretty even.

No, this isn't what's keeping me awake. I've reworked the instructions
as a solo game for simplicity



I would ask for two clarifications of the rules:
a) I would have thought that a player with three open squares then who
rolls a 1 or 2 would fill in the proper number of squares.

Yes, with three open squares, you have a 1/6 chance of winning (Throw
a 3), a 3/6 chance of losing (Throw a 4,5,6) and a 2/6 chance of
staying in to throw again (Throw a 1 or 2)

Grace's Dad

Let p[n] be the probability of winning if there are n open squares.
Then p[0] = 1, and p[n] = (sum p[n-i], i = 1,...,min[n,6])/6.
Here they are for n = 1,...,100. The maximum is at n = 6.

1 .16666667 26 .28574771 51 .28571426 76 .28571429
2 .19444444 27 .28576882 52 .28571427 77 .28571429
3 .22685185 28 .28573563 53 .28571429 78 .28571429
4 .26466049 29 .28570095 54 .28571430 79 .28571429
5 .30877058 30 .28569183 55 .28571429 80 .28571429
6 .36023234 31 .28570747 56 .28571428 81 .28571429
7 .25360440 32 .28572540 57 .28571428 82 .28571429
8 .26809402 33 .28572168 58 .28571428 83 .28571429
9 .28036895 34 .28571383 59 .28571429 84 .28571429
10 .28928846 35 .28571019 60 .28571429 85 .28571429
11 .29339312 36 .28571173 61 .28571429 86 .28571429
12 .29083021 37 .28571505 62 .28571428 87 .28571429
13 .27926319 38 .28571632 63 .28571429 88 .28571429
14 .28353966 39 .28571480 64 .28571429 89 .28571429
15 .28611393 40 .28571365 65 .28571429 90 .28571429
16 .28707143 41 .28571362 66 .28571429 91 .28571429
17 .28670192 42 .28571420 67 .28571429 92 .28571429
18 .28558673 43 .28571461 68 .28571429 93 .28571429
19 .28471281 44 .28571453 69 .28571429 94 .28571429
20 .28562108 45 .28571424 70 .28571429 95 .28571429
21 .28596798 46 .28571414 71 .28571429 96 .28571429
22 .28594366 47 .28571422 72 .28571429 97 .28571429
23 .28575570 48 .28571432 73 .28571429 98 .28571429
24 .28559799 49 .28571434 74 .28571429 99 .28571429
25 .28559987 50 .28571430 75 .28571429 100 .28571429

.

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