Question

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

Answer 1

dont really get it

Answer 2

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

Answer 3

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

Answer 4

oh woops

Answer 5

i thought it said 'any'

Answer 6

do you get the question now?

Answer 7

yes

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

notice
3 dark 17 non-dark is impossible

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

can i get a yes?

Answer 18

sure, i guess

Answer 19

lol

Answer 20

the explanation seems reasonable enough

Answer 21

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?

Answer 22

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 )

Answer 23

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

Answer 24

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) + ...

Answer 25

btw, what level of maths is this?

Answer 26

not sure

Answer 27

i meant like are you still in school or university?

Answer 28

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

Answer 29

im in university

Answer 30

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

Answer 31

i find this problem confusing

Answer 32

whoops, should have told you that at the start!

Answer 33

im trying to force these into mutually exclusive possibilities

Answer 34

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)

Answer 35

but are these equally likely?

Answer 36

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

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

what was your approach?

Answer 41

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

Answer 42

can you show me your work

Answer 43

i need your help :)