Question Paper of Probability/Statistics/Proof

Question 1:

$A c i r c u l a r f i l t e r p a p e r o f r a d i u s 6 c m i s c u t a n d f o l d e d t o m a k e a c o n i c a l f i l t e r . S h o w t h a t t h e v o l u m e, V, o f t h e c o n e i s \frac{1}{3} π r^{2} \sqrt{36 - r^{2}} w h e r e r i s t h e b a s e r a d i u s . S h o w t h a t \frac{d V}{d r} = \frac{2 π r \sqrt{36 - r^{2}}}{3} - \frac{π r^{3}}{3 \sqrt{36 - r^{2}}} F i n d t h e m a x i m u m v o l u m e o f t h e c o n e a n d t h e c o r r e s p o n d i n g r a d i u s i n e x a c t f o r m .$

Question 2:

$A b o x c o n t a i n s e v e n c a r d s, w i t h e a c h c a r d l a b e l l e d w i t h o n e o f t h e f o l l o w i n g n u m b e r s 0, 5, 5, 9, 9, 9 A p e r s o n d r a w s o n e c a r d s a t r a n d o m f r o m t h e b o x, t h e t h e n d r a w s a s e c o n d c a r d a t r a n d o m w i t n o u t r e p l a c i n g t h e f i r s t c a r d . i) W h a t i s t h e p r o b a l i l i t y t h a t t h e p e r s o n d r a w s a 9, t h e n a 5 ? i i) W h a t i s t h e p r o b a b l i t y t h a t t h e s u m o f t h e t w o n u m b e r s d r a w s i s a t m o s t 10 ? i i i) W h a t i s t h e p r o b a b i l t y t h a t t h e s e c o n d c a r d s d r a w s i s l a b e l l e d 5 ?$

Question 3:

$W h i c h e x p r e s s i o n i s u s e d t o f i n d t h e i n t e r e s t e a r n e d w h e n I P i s i n v e s t e d f o r n y e a r s a n d i n t e r e s t o f 10 % p . a . i s c o m p o u n d e d t w i c e y e a r l y ?$

$(A) l = P {(1.1)}^{n} — P (B) I = P (1.05)^{n} - P (C) l = P {(1.0 5)}^{n} — P (D) l = P {(1.05)}^{\frac{n}{2}} — p$

Question 4:

A referendum was held in the city of Toowoomba and the people asked if they would consider having recycled water for drinking. 40% were in favour of the proposal, while 60% were against it. If two people were chosen at random, what is the probability that

both agree to use recycled water for drinking?

one person is against using recycled water for drinking?

Question 5:

A student decides to save money over one year. In her first week, she puts aside $0.10. In the second week,

$0.40, then in the third week, $0.70, and so on with constant increases over time.
What amount will she deposit in her 52nd week?
How much has she saved altogether over the year?

Question 6:

$J o s e p h i n e b o r r o w s $ 300 000 t o b u y a u n i t . I n t e r e s t i s c a l c u l a t e d m o n t h l y a t t h e r a t e o f 6 % p e r a n n u m c o m p o u n d e d m o n t h l y . S h e a g r e e s t o r e p a y t h e l o a n w i t h e q u a l m o n t h l y I n s t a l m e n t s o f M a t t h e e n d o f e a c h m o n t h f o r 20 y e a r s . L e t A, b e t h e a m o u n t o w i n g a f t e r n m o n t h s . F i n d a n e x p r e s s i o n f o r A_{1} S h o w t h a t A_{2} = 300000 (1.005) - M (1.005) - M . F i n d t h e a m o u n t o f t h e m o n t h l y r e p a y m e n t (t o n e a r e s t 5 c e n t s) .$

Question 7:

For the integers 1 to 25, one integer is chosen at the random. What is the probability that it is:

Divisible by 5 and greater than 18.

Question 8:

$From the integers 1 to 25, one integer is chosen at random. What is the probability that it is:$

Divisible by 5 or greater than 18.

Question 9:

$The number of seats in each row of a theatre increases by 4 as you go from the front row to the back row. (i) If there are fifteen seats in the front row, show that there are (4 n + 11) seats in the n th row. (ii) If the theatre has 18 rows of seats, calculate the total number of seats in the theatre.$

Question 10:

$A survey shows that if Australian voters were asked to choose who they preferred as Prime Minister lulia Tillard or Tonv labhot 45 % would choose Julia Tillard, 40 % would choose Tony Jabbot and the remainder would be undecided. If two voters were chosen at random and asked to make this choice: (i) Find the probability that they would both be undecided. (ii) Find the probability that at least one would be undecided.$