## perl Group Title A box contains 8 dark chocolates, 8 white chocolates, and 8 milk chocolates. I choose chocolates at random (yes, without replacement; I’m eating them). What is the chance that I have chosen 20 chocolates and still haven’t got all the dark ones? what is your answer to that one year ago one year ago

1. KABRIC Group Title

dont really get it

2. UsukiDoll Group Title

choose milk chocolates choose white chocolates and just 4 dark chocolates XD

3. Chelsea04 Group Title

the question says that you still haven't got ALL the dark ones, so you COULD get 4 dark chocolates to 7 dark chocolates, but not the full 8

4. perl Group Title

oh woops

5. perl Group Title

i thought it said 'any'

6. Chelsea04 Group Title

do you get the question now?

7. perl Group Title

yes

8. UsukiDoll Group Title

I'm thinking milk white milk white milk white milk white milk white

9. perl Group Title

you wantthe probability of not getting all 8 dark chocolates. you are guaranteed at least 4 dark chocolates so the possibilities are 4 dark, 5 dark, 6 dark, 7 dark

10. Chelsea04 Group Title

yep, so if you find the probability of getting 8 dark chocolates, you can minus it from 1 and you'll get the probability of getting 4 dark, ... 7 dark

11. perl Group Title

there are 4 favorable cases, out of 5 total possible cases favorable cases : : exactly 4 dark , 16 non-dark : exactly 5 dark 15 non-dark exactly 6 dark 14 non-dark exactly 7 dark 13 non-dark total cases : exactly 4 dark , 16 non-dark : exactly 5 dark 15 non-dark exactly 6 dark 14 non-dark exactly 7 dark 13 non-dark exactly 8 dark , 12 non-dark I see no reason not to treat these choices as equally likely. therefore the probability of not picking all dark ones is 4/5

12. perl Group Title

since they must add up to 20 , these are the only possibilites

13. perl Group Title

notice 3 dark 17 non-dark is impossible

14. perl Group Title

the situation is constrained by two facts there must be no more than 16 non-dark chocolates there must be at least 4 dark chocalates

15. perl Group Title

that produces (16,4) (15, 5 ) (14,6) (13, 7) (12,8)

16. perl Group Title

the situation is constrained by two facts you cannot have more than 16 non-dark chocolates you cannot have more than 8 dark chocolates there must be at least 4 dark chocalates so the possibilities are (# non-dark, # dark) (16,4) (15, 5 ) (14,6) (13, 7) (12,8) there is 4/5 favorable

17. perl Group Title

can i get a yes?

18. Chelsea04 Group Title

sure, i guess

19. perl Group Title

lol

20. Chelsea04 Group Title

the explanation seems reasonable enough

21. perl Group Title

i think its more complicated, let X = number of dark ones , Y = number of non-dark chocolate how many ways can you choose 16 non dark and 4 dark? how many ways can you choose 15 non-dark and 5 dark?

22. perl Group Title

P( chance not getting all dark in 20 draws) = P( X = 4 & Y= 16 or X= 5 & Y=15 or X=6 & Y=14 or X=7 or Y=13 )

23. Chelsea04 Group Title

that's what i thought too, but if you add up all the ways to not get all 8 dark chocolate, i'm like 90% sure you'll get 4/5...cause there are like 60 ways if you account for each individual selection

24. perl Group Title

P( X = 4 & Y= 16 or X= 5 & Y=15 or X=6 & Y=14 or X=7 or Y=13 ) = P ( X=4 & Y=16) + P(X=5 & Y=15) + ...

25. Chelsea04 Group Title

btw, what level of maths is this?

26. perl Group Title

not sure

27. Chelsea04 Group Title

i meant like are you still in school or university?

28. perl Group Title

P(Y=16 & X=4) = P(Y=16) * P (X=4 | Y = 16) =

29. perl Group Title

im in university

30. Chelsea04 Group Title

oh, then you might not want to listen to me, i'm in yr 11 doing yr 12 maths so yea...

31. perl Group Title

i find this problem confusing

32. Chelsea04 Group Title

whoops, should have told you that at the start!

33. perl Group Title

im trying to force these into mutually exclusive possibilities

34. perl Group Title

you cannot have more than 16 non-dark chocolates you cannot have more than 8 dark chocolates there must be at least 4 dark chocalates so the possibilities are (# non-dark, # dark) (16,4) (15, 5 ) (14,6) (13, 7) (12,8)

35. perl Group Title

but are these equally likely?

36. perl Group Title

if these possibilities are equally likely to occur (and they are mutually exclusive), then the probability is 4/5

37. Chelsea04 Group Title

i'd assume so, unless some were heavier or bigger than the others (which isn't so)

38. perl Group Title

im use to working on problems like 5/12*4/11 , those sorts of problems

39. Chelsea04 Group Title

me too, I guess you could do that, but you'd be there forever!

40. perl Group Title

41. Chelsea04 Group Title

I found the probability of getting all 8 chocolates, kinda took a while and subtracted it from 1

42. perl Group Title

can you show me your work

43. perl Group Title

44. perl Group Title

|dw:1366635152002:dw|

45. Chelsea04 Group Title

hmm..i guess you could use combinatorics

46. Chelsea04 Group Title

does order matter?

47. perl Group Title

because i want Probability ( 4 dark& 16 non-dark) exactly

48. perl Group Title

$\frac{ 8C4*16C16 }{24C20 }+\frac{ 8C5*16C15 }{24C20}+\frac{ 8C6*16C14}{24C20}+\frac{ 8C7*16C13 }{24C20 }$

49. Chelsea04 Group Title

what do you get from that?

50. perl Group Title

the first is the case 4 dark & 16 non-dark then 5 dark& 15 nondark

51. perl Group Title

the order of eating them does not count, but we should label in our mind the chocolates. if you want, label the dark chocolates 1-8, and label the non-dark chocolates 9-24

52. perl Group Title

so now we have the problem dark chocolate = { 1,2,3,4,5,6,7,8} , non-dark { 9,10,11,12,... 23,24} , the numbers are going to be labels for the chocolates, ok ?

53. perl Group Title

i could have used letters,

54. Chelsea04 Group Title

sure

55. perl Group Title

so for instance, suppose we want four dark chocolates, and 16 non-dark chocolate. so we do probability = # favorable outcomes / # total possibilities the favorable, first choose 4 dark chocolates which has a total of 8 choose 4 ways to do it, then multiply by that how many ways can you choose 16 non-dark from 16 non dark. there is only one way. so im using multiplication rule (and probability)

56. perl Group Title

the denominator is the total number of ways to choose 20 chocolates , and there are 24 choose 20 ways to choose 20 chocolates

57. perl Group Title

anyways, show me your work, and we can compare answers . sorry if my explanation is confusing, im not exactly sure of the correct jargon

58. perl Group Title

so I got a probability of 629/759 = .8287

59. Chelsea04 Group Title

I tried using permutations so like: $\frac{ nPr(16,16)*nPr(8,4)+nPr(16,15)*nPr(8,5)...etc }{ nPr(24,20) }$ but then the amswer was: $\frac{ 5 }{ 245157 }$ which doesn't seem right...AT ALL... so i replaced all the P with C to use combinatorics and i got 629/759, which seems more accurate than permutations

60. Chelsea04 Group Title

it's close to 4/5 at least

61. perl Group Title

thats right, and yes its close to 4/5, but 4/5 would give you a wrong answer. i know how teachers are, lol

62. KABRIC Group Title

63. perl Group Title

and you could solve it using the complement approach .

64. perl Group Title

kabric, how do you know?

65. Chelsea04 Group Title

yea, basically what i was thinking

66. perl Group Title

chelsea, how did you get nCr using pretty print?

67. Chelsea04 Group Title

i just typed it?

68. perl Group Title

$1-\frac{ \left(\begin{matrix}8 \\ 8\end{matrix}\right)*\left(\begin{matrix}16 \\ 12\end{matrix}\right) }{ \left(\begin{matrix}24 \\ 20\end{matrix}\right) }$

69. perl Group Title

that is 8C8 , 16 C 12

70. perl Group Title

do you how it comes out the same ?

71. Chelsea04 Group Title

what do you mean?

72. Chelsea04 Group Title

you sort of lost me there

73. perl Group Title

oh ok sorry

74. perl Group Title

|dw:1366637463112:dw|

75. Chelsea04 Group Title

yes...

76. perl Group Title

P( X>7 ) = P(X=8 or X=9 or .. ) but X=9 and higher is impossible so we only need to look at X = 8

77. perl Group Title

we cant have 9 dark ones, since there arent 9 dark ones

78. Chelsea04 Group Title

sure

79. perl Group Title

ok lets not pursue this further, i have a new question answering, follow me

80. WalterDe Group Title