# Thread: Probability of receiving 8 pairs of identical boxes choosing from 8 items 16 times?

1. ## Probability of receiving 8 pairs of identical boxes choosing from 8 items 16 times?

A friend stated they bought 16 crates, each crate can contain 1 of 8 unique boxes. He opened the crates and received exactly 2 of each unique box.

2x box 1
2x box 2
2x box 3
2x box 4
2x box 5
2x box 6
2x box 7
2x box 8

I found that odd since the unique boxes in the crates were random. It seemed to suggest a pattern not randomness. I explained he could have received all of the same box, or none of the same box, or any mix in between.

What is the probability of receiving the above set of unique boxes from only 16 crates?

2. ## Re: Probability of receiving 8 pairs of identical boxes choosing from 8 items 16 time

Originally Posted by Blue7
A friend stated they bought 16 crates, each crate can contain 1 of 8 unique boxes. He opened the crates and received exactly 2 of each unique box.
2x box 1
2x box 2
2x box 3
2x box 4
2x box 5
2x box 6
2x box 7
2x box 8
I found that odd since the unique boxes in the crates were random. It seemed to suggest a pattern not randomness. I explained he could have received all of the same box, or none of the same box, or any mix in between. What is the probability of receiving the above set of unique boxes from only 16 crates?
Your grammar is terrible: "a friend...they bought"? What in the world are you asking?
Come on "2 of each unique box." Does that make any sense? What does 'unique' mean?

3. ## Re: Probability of receiving 8 pairs of identical boxes choosing from 8 items 16 time

Thank you for reply, maybe this is more clear, if not please let me know:

1 crate contains a random item.

The random item inside can be one of the following b1, b2, b3, b4, b5, b6, b7, b8.

If 16 crates are opened what is the probability of receiving the following in no particular order:

2 of b1
2 of b2
2 of b3
2 of b4
2 of b5
2 of b6
2 of b7
2 of b8

Thanks.

4. ## Re: Probability of receiving 8 pairs of identical boxes choosing from 8 items 16 time

Originally Posted by Blue7
Thank you for reply, maybe this is more clear, if not please let me know:

1 crate contains a random item.

The random item inside can be one of the following b1, b2, b3, b4, b5, b6, b7, b8.

If 16 crates are opened what is the probability of receiving the following in no particular order:

2 of b1
2 of b2
2 of b3
2 of b4
2 of b5
2 of b6
2 of b7
2 of b8

Thanks.
As all boxes are equiprobable the probability of getting 2 pairs of each color comes down to the ratio
of how many ways that can occur to the total possible number of 16 choices.

The number of arrangements of 8 pairs of boxes is given by

$N_p = \prod \limits_{k=0}^{6}\dbinom{16-2k}{2} = 81729648000$

This just successively selects two slots for each color except for the last.

The total number of possible selections of boxes is

$N_t = 8^{16} = 281474976710656$

$p = \dfrac{N_p}{N_t} = \dfrac{638512875}{2199023255552}\approx 0.00029$