Question Paper of Probability/Statistics/Proof

Question 1:

From a standard deck of cards (4 cards of each kind, 13 of each suite 52 cards in total), 3 cards are drawn without replacement.

(i) Complete the probability tree below on your paper displaying of probability of drawing kings from a standard deck without replacement:

(ii) Find the probability of drawing 2 kings in the 3 draws.

100 goats were released on to an island at the start of 1850 to provide food for shipwrecked sailors. By 1855 there were 180 goats on the island. Assuming the goat population since 1850 to follow the rule $P = A e^{k t}$ where t is the time in years since 1850:

(i) Find the values of A and k.

(ii) Find the expected goat population in 1860.

(iii) Once there are more than 1000 goats on the islands they start to eat more vegetation than the island can regenerate. In what year did this occur?

Question 2:

$F i n d t h e l i m i t i n g s u m o f t h e g e o m e t r i c s e r i e s 6 - 4 + 2 \frac{2}{3} - . . .$

Question 3:

$A s c h o o l c o m m i t t e e c o n s i s t s o f f i v e y e a r 12 g i r l s, s i x y e a r 11 g i r l s a n d t w o y e a r 10 g i r l s . T w o g i r l s a r e c h o s e n a t r a n d o m f r o m t h i s c o m m i t t e e t o r e p r e s e n t t h e s c h o o l a t a f u n c t i o n . F i n d t h e p r o b a b i l i t y t h a t : t h e y a r e b o t h y e a r 12 g i r l s o n e y e a r 12 g i r l a n d o n e y e a r 11 g i r l i s c h o s e n a t l e a s t o n e y e a r 10 g i r l i s c h o s e n$

Question 4:

$I f m + 3 m + 5 m + . . . + 53 m = 81, f i n d t h e v a l u e o f m .$

Question 5:

$T h e d e p t h o f w a t e r i n t h e c r o s s - s e c t i o n o f a 4 m e t r e w i d e c r e e k w a s m e a s u r e d a n d r e c o r d e d i n t h e t a b l e b e l o w$

$B y a p p l y i n g S i m p s o n' s R u l e, f i n d t h e c r o s s - s e c t i o n a l a r e a o f t h e c r e e k a t t h i s p o i n t i n s q u a r e m e t r e s c o r r e c t t o 2 d e c i m a l p l a c e s . T h e w a t e r a t t h i s p o i n t i s f l o w i n g a t a r a t e o f 0.5 m / s . C a l c u l a t e t h e v o l u m e o f w a t e r w h i c h p a s s e s t h i s p o i n t i n o n e m i n u t e (a n s w e r t o t h e n e a r e s t c u b i c m e t r e) .$

Question 6:

$E v e r y y e a r, s t a r t i n g o n L a n a' s f i r s t b i r t h d a y, h e r g r a n d p a r e n t s d e p o s i t e d $ 100 f o r h e r i n a s p e c i a l b a n k a c c o u n t a t a r a t e o f 9 % p . a ., c o m p o u n d e d a n n u a l l y . O n h e r 21 " b i r t h d a y, i n s t e a d o f d e p o s i t i n g $ 100, t h e y d e p o s i t e d a l u m p s u m o f $ 10 000 i n t o t h e s a m e a c c o u n t . A f t e r t h i s, t h e y s t o p p e d d e p o s i t i n g m o n e y f o r L a n a . H o w m u c h d i d L a n a h a v e i n h e r a c c o u n t i m m e d i a t e l y a f t e r t h e l u m p s u m o f $ 10 000 w a s d e p o s i t e d ? L a n a l e f t a l l t h e m o n e y i n t h e b a n k a t t h e s a m e i n t e r e s t r a t e u n t i l h e r 26 t h b i r t h d a y . W h a t w a s t h e b a l a n c e o f h e r a c c o u n t t h e n ? W h a t \sin g l e a m o u n t o f m o n e y w o u l d L a n a' s g r a n d p a r e n t s h a v e n e e d e d t o i n v e s t o n h e r f i r s t b i r t h d a y s o t h a t s h e h a d t h e s a m e a m o u n t o f m o n e y o n h e r 26 t h b i r t h d a y ?$

Question 7:

$O z t o w n h a d a 25 y e a r h o u s e b u i l d i n g p r o g r a m s t a r t i n g a t t h e b e g i n n i n g o f 1991 a n d f i n i s h i n g a t t h e e n d o f 2015 . T h e n u m b e r o f h o u s e s b u i l t e a c h c a l e n d a r y e a r f o l l o w s a n a r i t h m e t i c p r o g r e s s i o n w i t h f i r s t t e r m a a n d c o m m o n d i f f e r e n c e d . 1900 h o u s e s w e r e b u i l t i n t h e y e a r 2000 a n d 1100 h o u s e s w e r e b u i l t i n t h e y e a r 2010 . i) F i n d t h e v a l u e s o f a a n d d . i i) F i n d t h e t o t a l n u m b e r o f h o u s e s b u i l t o v e r t h e 25 y e a r s .$

Question 8:

$J o s e b o r r o w e d $ 250 000 a t t h e b e g i n n i n g o f 2008 . T h e a n n u a l i n t e r e s t r a t e i s 8 % . A t t h e e n d o f e a c h y e a r, i n t e r e s t i s c a l c u l a t e d o n t h e b a l a n c e a t t h e b e g i n n i n g o f t h e a n d a d d e d t o t h a t b a l a n c e o w i n g T h e d e b t i s t o b e r e p a i d b y e q u a l a n n u a l r e p a y m e n t s o f $ 30 000, w i t h t h e f i r s t p a y m e n t b e i n g m a d e a t t h e e n d o f 2008 . S h o w t h a t A_{2} = 250 000 {(1.08)}^{2} - 30000 (1 + 1.08) S h o w t h a t A_{n} = 375000 - 125 000 {(1.08)}^{n} I n w h i c h y e a r w i l l J o s e m a k e t h e f i n a l r e p a y m e n t ?$

Question 9:

Explain why the series 1+(√2-1)+(√2-1)²+... has a limiting sum.

Calculate the limiting sum of this series $E x p l a i n w h y t h e s e r i e s 1 + (\sqrt 2 - 1) + (\sqrt 2 - 1) ² + . . . h a s a l i m i t i n g s u m . C a l c u l a t e t h e l i m i t i n g s u m o f t h i s s e r i e s$

Question 10:

$p i s a n i n t e g e r c h o s e n a t r a n d o m f r o m t h e s e t 5, 7, 9, 11 q i s a n i n t e g e r c h o s e n a t r a n d o m f r o m t h e s e t 2, 6, 10, 14, 18 W h a t i s t h e p r o b a b i l i t y t h a t p + q = 23 ?$