1.A force of 40.0 N is needed to compress a spring 0.200 m. A 1.00 x 10-2 kg ball
a) Calculate the work done to compress the spring. (2 marks)
b) What happens to the work done on the spring ? (1 mark)
c) If the spring is released, what happens to the energy of the spring? (1 mark)
d) Calculate the total mechanical energy of the ball at the instant it leaves the spring. (2 marks)
e) What will be the speed of the ball at the instant it leaves the spring? (2 marks)
f) If the ball is fired up into the air by the spring, how much gravitational potential energy will it gain? (1 mark)
g) What will be the maximum height of the ball? (2 marks
2.Exercise 4) A fair coin is tossed. If it lands heads, a fair four-sided die
is thrown (with values 2,3,4,7). If
2,3,4,7). If it lands tails, a fair six-sided die is
thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice
eats n grains of rice, where n is the largest prime factor of the die result
(for example, the largest prime factor of 9 is 3).
(a) What is the conditional probability that the coin lands heads, given
that Alice eats three grains of rice?
(b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through).
What is the conditional probability that the coin lands heads on both
run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs?
(Do not count the two grains from part (a) in part (b); we assume
two brand new experiments, each with a new coin toss. Start your
solution by defining a suitable partition of the sample space. Please
use an appropriate notation and/or justification in words, for each
value that you give as part of your solution.)
Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 .
Each time the unfair coin lands heads, Alice gives Bob £1 . Each time
the unfair coin lands tails, Bob gives Alice £1 . The game ends when one
player has £5 .
(a) Draw a labelled Markov chain describing the problem, and write
down a transition matrix P. Write down the communication classes,
and classify them as either recurrent or transient.
(b) Using the transition matrix, calculate the probability that Alice loses
all of her money in exactly four tosses of the unfair coin.
(c) Calculate the (total) probability that Alice loses all of her money
(before Bob loses all of his).
(d) Calculate the expected (mean) number of tosses of the unfair coin,
for the game to end.
3.We play a variation of Monty Hall: The contestant is presented with 3 curtains, behind each there's a prize with
a prize with probability p independently of the other curtains. The player chooses one curtain
(a) The game host tells the contestant there's a prize hiding behind at least one of the other curtains, and offers the player a chance to change his choice. Should he?
(b) The game host tells the player there's a prize hiding behind exactly one of the other curtains. Should the player change his choice? How does your answer depend on p
(c) How would your answer to section (b) change if there are n curtains in total?
(d) Bonus (4 points): How would your answer to section (a) change if there are 4 curtains in total?
4.Let A be the set that is defined recursively in the following way:
Basis Case: ( 0 , 1 , 2
Recursive Step: if (n,m,r) ∈ A then (n+1,m+2,m+r+2) ∈ A
Prove, using structural induction, that for every .( n , m , r ) ∈ A , 0 ≤ n < m and n + m ≤ r .
5.Fox News recently reported the results of a public opinion poll on supporting Trump that asked:
“Since he became the president,
Since he became the president, did President Trump act with the transparency and the integrity
that you expect from a president?” 675 voters responded the poll and 351 responded “YES.”
Assume that 40% of the U.S. population supports Trump.
a. Define a binary random variable, Y, for supporting Trump (Y=1) vs. not (Y=0).
Calculate the population mean (????????) and variance (????????
) for supporting Trump.
b. Calculate the sample mean ????̅ and the sample standard deviation of ????̅ (????????̅ ) for the poll.
c. Calculate the standard error of ????̅ and construct a 95% confidence interval from the
poll using ????̅ and its sample standard error.
d. Conduct a two-sided hypothesis test at 5% significance level to determine whether
40% of the U.S. population supports Trump. State the null and the alternative
hypotheses, calculate the test statistics and the associated p-value, and conclude. Is
the Fox News survey reliable? Why? Why Not?
e. Suppose that you wanted to design a survey that had a margin of error of at most 1%.
That is: the difference between the upper bound and the lower bound of the
confidence interval should be a maximum of 2 percentage points. For example, for
????̅ = 0.52 you are aiming for the 95 % CI to be [0.51 0.53].
How large should n be if the survey uses simple random sampling?
6.Suppose you measure a block’s weight by hanging it from a spring scale. You find that it
weighs 34.0 N
34.0 N when it’s not in the water. When it’s submerged in water (the density of water is
1.00 x 103 kg/m3) the scale now reads 27.0 N. (a) What is the density of the block? (b) If you
suspended another object from the block that has a density of 3.20 x 103 kg/m3, with both objects
submerged, what would the object's mass need to be for the scale to once again read 34.0 N?
Note: Part (a) is worth 7 points, and part (b) is worth 8 points.
7.Twenty students are asked to select an integer between 1 and 10. Eight choose either 4, 5 or 6.
f the students make their choices independently and each is as likely to pick one integer
as any other, what is the probability that 8 or more will select 4,5 or 6?
b Having observed eight students who selected 4, 5, or 6, what conclusion do you draw based
on your answer to part (a)?A missile protection system consists of n radar sets operating independently, each with a
probability of .9 of detecting a missile entering a zone that is covered by all of the units.
a If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect
the missile? At least one set?
b How large must n be if we require that the probability of detecting a missile that enters the
zone be .999?
8.A mathematician diagnosed with schizophrenia is fooling himself by playing
with a list L containing n distinct integers in just his
ng n distinct integers in just his thoughts.
He plays turns on the list. In each turn he does the following–
He takes a number (in its index’s order ) and swap it with any number in
the list including itself i.e. if it swap it with itself it doesn’t move at all (The
selection of the number is completely random).
He does the same for all the elements in their index’s order in that turn.
If initially the list was unsorted, such that, no element was in sorted position,
then find the probability that the list is sorted after m such turns.
Note : Take the assumption that if an element is not in its sorted
position then it can be in any other n − 1 positions equally likely.