Question

I am with a group of five of my friends. A box contains 8 pieces each of milk chocolate, white chocolate, and dark chocolate. The box is passed around the six of us, with each person taking 4 pieces. Assume that each person chooses at random without replacement from the available pieces. I am the last person to whom the box is passed. Find the chance that I pick 4 dark chocolates.

Answer 1

Ok, let me give you some ideas. Perhaps you will be able to figure it out.

It doesn't matter whether I am the last person or the first. Since each person is an equal element, the probability that any one picks 4 dark chocolates is the same. Also the probability of picking 4 dark chocolates in the first pick is same as picking 4 dark chocolates in the last pick.

The logic behind it is a little un-intuitive but I have discussed it in detail in this post. Check it out:
http://www.veritasprep.com/blog/2012/10 ... ure-again/

Now we need to find the probability of picking 4 dark chocolates in the first pick.

Find the total number of ways in which you can pick 4 chocolates:
all 4 same type,
3 same type 1 different,
2 same 2 same,
2 same 2 different

Answer 2

Hey nathan...
I've tried this method before posting the question itself....
It didn't work actually....
If we try in this way we get 1/12..

Answer 3

Actually, I'm doing the statistics course in EDX...
It shows that my answer is wrong...
Do you have any other methods???

Answer 4

No not Really this question is confusing.

Answer 5

But ill try to find more methods.

Answer 6

Possible Solution: The probability of picking 4 dark chocolates in the first pick is same as picking 4 dark chocolates in the last pick" the answer should be: 8/24 * 7/23 * 6/22 * 5/21 = 6.58E^-0.3

How about this?

Answer 7

@RadEn What do you think?

Answer 8

@narahari93 What do you think?

Answer 9

"The probability of picking 4 dark chocolates in the first pick is same as picking 4 dark chocolates in the last pick"...
This intuition is a bit confusing...

Answer 10

Coz, the last pick depends on the previous 5 picks....

Answer 11

we have to find the chance i pick 4 dark chocolates in a row is the same as drawing 4 cards with the same suit (try thinking the chocolates as a deck of cards). so the chance is 8/24*7/23*6/22*5/21. the fact that i draw last does not change the probability because the previous picks are unconditional to my picks.

Answer 12

That's another solution.

Answer 13

Ah so You agree :P.

Thank you for medal.

Answer 14

As I am the last one to pick, my probabilities of running into 4 dark chocolates left rather than picking them is the probability that in the first 20 picks (my friends´) they have picked already four dark chocolates. This is hypergeometric distribtion and formula is:
\[\frac{ \left(\begin{matrix}8 \\ 4\end{matrix}\right) \left(\begin{matrix}24-8 \\ 20-4\end{matrix}\right)}{ \left(\begin{matrix}24 \\ 20\end{matrix}\right) }=\frac{ 8·7·6·5 }{ 24·23·22·21 }\]