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


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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

$H i n t : T h e r e s u l t s o f t h a t e x p e r i e n c e$ $a r e s e t s {(A, B) / A, B \subset {1, . . . ., n}}$ $o f q u e s t i o n s c o r r e c t l y a n s w e r e d$ $A n d t h e e v e n t h e r e a r e a l l t h o s e$ $s u c h a s A \cap B = \emptyset$
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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