Question Paper of Probability/Statistics/Proof

Question 1:

$T h e f i r s t t e r m o f g e o m a t r i c s e r i s e i s 16 a n d t h e c o m m o n r a t i o i s \frac{1}{n} . i) F o r w h a t v a l u e n w i l l t h i s s e r i s e h a v e l i m i t u n g s u m ? i i) C a l c u l a t e t h e l i m i t i n g s u m o f t h e s e r i s e w h e r e n = 4$

Question 2:

$T h e f i r s t 3 t e r m s o f a n A r i t h m e t i c P r o g r e e s i o n s a r e 50, 43, 36 . I f t h e l a s t t e r m i s - 27, f i n d t h e s u m o f t h e s e r i s e s .$

Question 3:

chance of heavy traffic is If it is not a rainy day, the chance of heavy traffic is 1 4 If it Neville lives in a town where one-third of the days are rainy days. If it is a rainy day, the is a rainy day and there is heavy traffic, there is a 50% chance that Neville will arrive late to work. On the other hand, the probability of being late is reduced to if it is not a rainy day and there is no heavy traffic. In other situations, the probability of Neville being late to work is 0.35. On a random day, Neville arrives late to work. What is the probability that it is a rainy day!

Question 4:

A geometric series with a common ratio r will have a limiting sum if

$(A) r < 1 (B) r > 1 (C) |r| < 1 (D) |r| > 1$

Question 5:

$j o a n d e p o s i t $ 350 i n t o a s p e c i a l s a v i n g s a c o u n t o n t h e f i r s t d a y o f e a c h m o n t h f o r t w o y e a r s . t h e i n t e r e s t r a t e i s 9 % p . a . c o m p o n d e d m o n t h l y . f i n d t h e t o t a l a m o u n t i n h e r s a v i n g s a c o u n t s a t t h e e n d o f t h e t w o y e a r p e r i o u d .$

Question 6:

$A S u m o f $ 15000 i s b o r r o w e d a t 15 % p a i n t e r e s t, c a l c u l a t e o n t h e b a l a n c e o w i n g a t t h e e n d o f e a c h m o n t h . T h e m o n e y i s t o r e p a i d a t m o n t h l y i n t e r v a l s o v e r 5 y e a r s i) I f M i s s \tan d s f o r m o n t h l y r e p a y m e n t, s h o w t h a t t h e a m o u n t o w i n g a t t h e e n d o f 2 n d m o n t h i s g i v e n b y A_{2} = 15000 {(1.01)}^{2} - M (1.01 + 1) i i) W r i t e a g e n e r a l e x p r e s s i o n f p r 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 i i i) f i n d a m o n t h l y r e p a y m e n t$

Question 7:

$F i n d t h e v a l u e o f n w h i c h w h e n a d d e d t o e a c h 2, 5, a n d 9 w i l l g i v e t h r e e n u m b e r s i n g e o m a t r i c p r o g r e s s i o n s$

Question 8:

Evaluate the arithmetic series 3 +7 +11 +15 +··· +4003.

Question 9:

A bag contains 8 blue marbles and 4 yellow marbles. Two marbles are selected at random without replacement.

(i) Draw a tree diagram to show all possible outcomes. Include the probability on each branch.

(ii) What is the probability that the two marbles are of different colours?

Question 10:

On 1st July 2015, Jessica invested $18 000 in a bank account that paid interest at a rate of 5% p.a, compounded annually.

(i) How much would be in the account after the payment of interest on 1st July 2025 if no additional deposits were made?

(ii) Consider if Jessica made additional deposits of $1500 to her account on the 1st July each year, beginning on 1st July 2016. After the payment of interest and her deposit on 1st July 2025, how much was in her account?