I-have-n-six-sided-dice-I-roll-them-all-What-is-the-proability-that-k-of-them-share-the-same-value-

Question Number 3894 by Filup last updated on 24/Dec/15

I have n six sided dice .

I roll them all . What is the proability

that k of them share the same value ?

Commented by Filup last updated on 24/Dec/15

Whoops . That was my mistake!

Commented by Filup last updated on 24/Dec/15

P (1 dice) = \frac{1}{6}

\Rightarrow P (n - dice) = \frac{1}{6^{n}}

No . of combinations (n o d u p l i c a t e s)

=^{n} C_{k} = \frac{n!}{(n - k)!}

Commented by Yozzii last updated on 24/Dec/15

S o y o u h a v e a t m o s t s i x d i c e ?

Commented by Filup last updated on 24/Dec/15

You have n dice . Each dice has 6 sides .

Commented by Yozzii last updated on 24/Dec/15

Y o u s h o u l d t h e n o m i t t h e n o t e t h a t

1 ⩽ k ⩽ n ⩽ 6, u n l e s s 1 ⩽ k ⩽ 6 . T h e n, j u s t

r e m o v e n ⩽ 6 .

Answered by prakash jain last updated on 24/Dec/15

Total number of outcomes = 6^{n}

Favorable outcome = 6 \times^{n} C_{k} \times 5^{n - k}

k dices to give the same result .

^{n} C_{k} dices to give that result

(n - k) dices can have any of other 5 results

probability = \frac{^{n} C_{k} \times 5^{n - k}}{6^{n - 1}}

Commented by Yozzii last updated on 25/Dec/15

W h a t i f n = 2 k ? T h e n k c h o s e n d i c e s c o u l d

h a v e t h e s a m e r e s u l t, e . g 6, w h i l e

t h e o t h e r k d i c e s c o u l d p o s s i b l y h a v e k i d e n t i c a l

r e s u l t s, e . g 1, 2, 3, 4, 5 .

p r o b a b i l i t y = \frac{(\begin{matrix} 2 k \\ k \end{matrix}) 5^{k} - 5}{6^{2 k - 1}} ?

Commented by prakash jain last updated on 25/Dec/15

Good point . The above answer is incorrect .

Commented by Yozzii last updated on 25/Dec/15

Y e s . I t a p p e a r s a b i t t r i c k y . T h e c a s e s

c o n c e r n i n g n a n d k m i g h t b e n = k, k < n < 2 k,

n = 2 k, n > 2 k, n = m k, m \in Z^{+} .

Commented by prakash jain last updated on 25/Dec/15

I just considerred the case where n < 2 k .