2-students-are-passing-a-test-of-n-questions-with-the-same-chance-to-find-each-one-Show-the-chance-that-they-both-don-t-find-a-same-question-is-3-4-n-

Question Number 207372 by sniper237 last updated on 12/May/24

2 s t u d e n t s a r e p a s s i n g

a t e s t o f n q u e s t i o n s w i t h

t h e s a m e c h a n c e t o f i n d e a c h o n e

S h o w t h e c h a n c e t h a t t h e y b o t h

{d o n}^{'} t f i n d a s a m e q u e s t i o n i s {(\frac{3}{4})}^{n}

Commented by A5T last updated on 13/May/24

I f t h e c h a n c e o f f i n d i n g (g e t t i n g ?) o n e q u e s t i o n

i s \frac{x}{y} ⩽ 1; a n d t h e s t u d e n t s b e A a n d B . W h e n n = 1;

t h e r e a r e t h r e e c a s e s :

(i) A g e t s t h e q u e s t i o n a n d B m i s s e s .

(i i) B g e t s t h e q u e s t i o n a n d A m i s s e s

(i i i) A a n d B m i s s t h e q u e s t i o n

\Rightarrow P r o b a b i l i t y = \frac{x}{y} (1 - \frac{x}{y}) + (1 - \frac{x}{y}) (\frac{x}{y}) + \frac{{(y - x)}^{2}}{y^{2}}

= \frac{x (y - x)}{y^{2}} + \frac{{(y - x)}^{2}}{y^{2}} = \frac{(y - x) (y - x + x)}{y^{2}} = \frac{y - x}{y}

S o, t h e p r o b a b i l i t y i s d e p e n d e n t o n \frac{x}{y} .

F o r e x a m p l e, i f \frac{x}{y} = \frac{1}{4}; t h e n p r o b a b i l i t y = \frac{3}{4} .

B u t i f \frac{x}{y} = 1; t h e n p r o b a b i l i t y = 0

Commented by sniper237 last updated on 13/May/24

H i n t : T h e r e s u l t s o f t h a t e x p e r i e n c e

a r e s e t s {(A, B) / A, B \subset {1, \dots ., n}}

o f q u e s t i o n s c o r r e c t l y a n s w e r e d

A n d t h e e v e n t h e r e a r e a l l t h o s e

s u c h a s A \cap B = \emptyset

Commented by A5T last updated on 13/May/24

I t i s d e p e n d e n t o n \frac{x}{y} (t h e p r o b a b i l i t y o f t h e m

g e t t i n g t h e c o r r e c t a n s w e r) . S o, u n l e s s \frac{x}{y} = \frac{1}{4},

{(\frac{3}{4})}^{n} f a i l e d t o h o l d f o r n = 1 .

Commented by sniper237 last updated on 13/May/24

R e a d t h e p o s t a g a i n . y o u {d o n}^{'} t n e e d t h a t

Commented by A5T last updated on 13/May/24

D o y o u h a v e a s o l u t i o n ?

Commented by A5T last updated on 13/May/24

(i) A g e t s t h e q u e s t i o n a n d B m i s s e s .

(i i) B g e t s t h e q u e s t i o n a n d A m i s s e s

(i i i) A a n d B m i s s t h e q u e s t i o n

(i v) A a n d B g e t t h e q u e s t i o n

T h e n t h e r e a r e \frac{3}{4} w a y s f o r A a n d B n o t t o g e t

t h e s a m e q u e s t i o n ?

S o, {(\frac{3}{4})}^{n} f o r n q u e s t i o n s ?

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