Am-not-a-friend-with-this-isssue-6-red-1-black-and-3-white-balls-We-draw-3-balls-in-a-raw-returning-the-drawn-ball-each-time-How-many-different-results-which-include-at-least-one-black-ball-W

Question Number 178134 by Acem last updated on 13/Oct/22

A m n o t a f r i e n d w i t h t h i s i s s s u e :

6 r e d, 1 b l a c k a n d 3 w h i t e b a l l s

W e d r a w 3 b a l l s i n a r a w, r e t u r n i n g t h e

d r a w n b a l l e a c h t i m e .

H o w m a n y d i f f e r e n t r e s u l t s w h i c h i n c l u d e

a t l e a s t o n e b l a c k b a l l .

{W a y}_{1} : 1000 - 9^{3} = 271 r e s u l t s {T h a t}^{'} s v e r y o k

{W a y}_{2} : 1 \times 9 \times 9_{v e r y o k} \times 3_{O m g, W h y ?}

+ 1 \times 1 \times 9 \times 3_{t h e s a m e c o n f u s e d a b o v e}

+ 1 \times 1 \times 1_{v e r y o k a n d i l i k e i t \dots . b u t w h y n o t \times 3}

= 271 r e s u l t

{E x p l a n a t i o n}_{a b o u t t h e s t o r y o f \times 3} ?

Commented by Acem last updated on 13/Oct/22

H e a s k e d m e a b o u t f i n a l g e t t i n g 1 b l a c k b a l l .

I d o n e! t h e q u e s t i o n {d o e s n}^{'} t c a r e

o f a r r a n g i n g t h i n g s .

A n y w a y g i v e m e s o m e t i m e t i l l F r e o n 13

s e t t l e i n m y b r a i n t a n k .

Commented by mr W last updated on 13/Oct/22

E x p l a n a t i o n t o W a y 2

c a s e 1 : o n e b l a c k b a l l, t w o b a l l s f r o m

o t h e r c o l o r s

1 \times 9 \times 9

s i n c e t h e b l a c k b a l l c a n b e t h e f i r s t o n e,

t h e s e c o n d o n e o r t h e t h i r d o n e,

t h e r e f o r e t o t a l l y 1 \times 9 \times 9 \times 3 p o s s i b i l i t i e s

c a s e 2 : t w o b l a c k b a l l s a n d o n e b a l l

f r o m o t h e r c o l o r s

1 \times 1 \times 9

s i n c e t h e b l a c k b a l l s c a n b e t h e f i r s t t w o,

t h e l a s t t w o o r t h e f i r s t a n d t h e l a s t

o n e, t h e r e f o r e t o t a l l y 1 \times 1 \times 9 \times 3

p o s s i b i l i t i e s

c a s e 3 : t h r e e b l a c k b a l l s

1 \times 1 \times 1

a l l t o g e t h e r :

1 \times 9 \times 9 \times 3 + 1 \times 1 \times 9 \times 3 + 1 \times 1 \times 1 = 271

Commented by mr W last updated on 13/Oct/22

i f w e h a d 2 b l a c k b a l l s i n s t e a d o f o n e

b l a c k b a l l i n t h e b a g, t h e r e s u l t i s

2 \times 9 \times 9 \times 3 + 2 \times 2 \times 9 \times 3 + 2 \times 2 \times 2 = 602

w e c a n g e t t h i s r e s u l t a l s o f r o m

11^{3} - 9^{3} = 602

Commented by Tawa11 last updated on 13/Oct/22

Great sir

Commented by Acem last updated on 13/Oct/22

F i r s t o f a l l, t h a n k y o u f o r y o u r i n t e r e s t a n d

d e t a i l e d e x p l a n a t i o n .

A c c o r d i n g t o m a t h e m a t i c a l l o g i c, w e c a r e a b o u t

t h e o r d e r o f t h e e l e m e n t s o f t h e c h o s e n s e t .

F o r e x a m p l e c h o o s i n g l e t t e r s f r o m a g r o u p t o

f o r m a w o r d (d i f f . w o r d s b y c h a n g i n g t h e l e t t e r s),

c h o o s i n g n u m b e r s t o f o r m n u m b e r s (d i f f . n u m .),

c h o o s i n g p e o p l e t o a s s i g n t h e m

s p e c i f i c t a s k s (d i f f . {p e r s o n}^{'} s j o b) \dots e t c

I n a l l t h e a b o v e w e c o n c e r e d i n a l l

t h e i r p o s s i b i l i t y p e r m u t a t i o n s

S o h o w c a n y o u m a k e m e c o n v i n c e o r a c c e p t

m e n t a l l y o f t h e i m p o r t a n c e o f t h e o r d e r o f t h e

s e l e c t e d o n l y d i f f e r e n t c o l o r b a l l s :

∙ ∙ ∙, ∙ ∙ ∙ H e r e i s e e n o d i f f e r e n c e, I {c a n}^{'} t

s e e t h a t {t h e r e}^{'} s d i f f e r e n t j o b s, d i f f e r e n t p h r a s e s

n e i t h e r d i f f e r e n t n u m b e r s

o r e v e n d i f f e r e n t f o r m s .

T h e r e s u l t i s i g o t o n e, t w o . . e t c b l a c k b a l l s

A c c o r d i n g t o {w a y}_{1} : 10^{3} - 9^{3} = 271

i k n o w a m w r o n g b u t n o t i n a l l c a s e s w e c a n u s e

t h e {w a y}_{1} a n d t h a t t i m e w i l l u s e t h e g e n e r a l

p r i n c i p l e c o u n t i n g t h e n i^{'} m s u r e i^{'} l l m a k e

a m i s t a k e .

P l e a s e h e l p m e t o s e e t h a t t h e r e a r e d i f f e r e n c e s

i n t h e p e r m u t a t i o n s a t t h i s i s s u e .

A n d m a k e m e f r i e n d w i t h i t .

Commented by mr W last updated on 13/Oct/22

i c a n u n d e r s t a n d y o u d o u b t .

e v e n w h e n w e a r e n o t t a l k i n g a b o u t

p r o b a b i l i t y i n c u r r e n t q u e s t i o n, b u t

a c t u a l l y i t i s a b o u t p r o b a b i l i t y . t h e n

w e m u s t t r e a t t h e b a l l s w i t h t h e s a m e

c o l o r a s d i f f e r e n t o b j e c t s . e x a m p l e :

i n a b a g t h e r e a r e 5 r e d b a l l s a n d o n e

b l a c k b a l l . n o w y o u r a n d o m l y t a k e

t w o b a l l s . h o w m a n y p o s s i b i l i t i e s

f o r t a k i n g o n e r e d a n d o n e b l a c k b a l l ?

w e k n o w t h e r e a r e 5 p o s s i b i l i t i e s,

e v e n t h e y a l l l o o k l i k e t h e s a m e :

b l a c k + r e d . y o u {d o n}^{'} t c a r e w h i c h r e d

b a l l i s t a k e n, b e c a u s e t h e y a r e t h e

s a m e t o y o u . b u t t h e b a l l s t h e m s e l f

c a r e . t h e r e i s d i f f e r e n c e i f t h e b a l l

A i s t a k e n o r t h e b a l l B i s t a k e n .

Commented by mr W last updated on 13/Oct/22

f o r t h e s a m e r e a s o n i n t h e w o r l d o f

p r o b a b i l i t y

∙ ∙ ∙, ∙ ∙ ∙

a r e t w o d i f f e r e n t t h i n g s .

Commented by Acem last updated on 13/Oct/22

I b a c k t o t h e q u s e t i o n, i t a s k s a b o u t

d i f f e r e n t r e s u l t s

h m m m \dots . i n e e d a b i t t i m e t o s e e i f {t h e r e}^{'} s

a d i f f e r e n c e b e t w e e n {\begin{cases} h o w m a n y r e s u l t \\ h o w m a n y d i f f e r e n t r e s . \end{cases}

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