4.5. Exercises

1. Use a for statement to print 10 random numbers.

   import random howmany = 10 for counter in range(howmany): arandom = random.random() print(arandom)

2. Repeat the above exercise but this time print 10 random numbers between 25 and 35.

4.5.1. Contributed Exercises

Q-1: After completing the reading, what concepts are still unclear to you? If nothing is unclear, what did you find most interesting?

Q-1: (TEST) The correct code to generate a random number between 1 and 100 (inclusive) is:

• prob = random.randrange(1, 101)
• This will generate a number between 1 and 101, but does not include 101.
• prob = random.randrange(1, 100)
• This will generate a number between 1 and 100, but does not include 100. The highest value generated will be 99.
• prob = random.randrange(0, 101)
• This will generate a number between 0 and 100. The lowest value generated is 0. The highest value generated will be 100.
• prob = random.randrange(0, 100)
• This will generate a number between 0 and 100, but does not include 100. The lowest value generated is 0 and the highest value generated will be 99.

In statistical physics, it is common to use Stirling’s approximation for \(N!\),

\[N!\approx N^N e^{-N} \sqrt{2\pi N}.\]

Obtain an integer from the user, assign it to N, and print out, in order, \(N!\), Stirling’s approximation, and the relative percent error. The relative percent error is calculated as

\[\text{error} = \left| \frac{\text{approx} - \text{exact}}{\text{exact}} \right| 100\%.\]

==== from unittest.gui import TestCaseGui import re class myTests(TestCaseGui): def testOne(self): ref_N_fact = math.factorial(N) ref_stirling = N**N * math.exp(-N) * math.sqrt(2 * math.pi * N) ref_rel_err = abs( (ref_stirling - ref_N_fact)/ref_N_fact) * 100 output = self.getOutput().split('\n') editor = self.getEditorText().split('\n') self.assertAlmostEqual(float(output[0]), ref_N_fact, 7, 'Checking N!') self.assertAlmostEqual(float(output[1]), ref_stirling, 5, 'Checking Stirling Approximation') self.assertAlmostEqual(float(output[2]), ref_rel_err, 7, 'Checking the relative error') self.assertFalse(re.search(r'( *import *math *as| *from *math)', '\n'.join(editor)), 'Checking import statement') myTests().main()

For a right triangle we have

\[\tan(\theta) = \frac{\text{opp}}{{adj}}\]

Compute and print \(\theta\) for

\[\begin{align} \left(\text{opp}=1,\text{adj}=1\right), \left(\text{opp}=1,\text{adj}=-1\right), \left(\text{opp}=-1,\text{adj}=1\right),\text{and} \left(\text{opp}=-1,\text{adj}=-1\right). \end{align}\]

Hint: use the function atan2() instead of atan().

==== from unittest.gui import TestCaseGui import re class myTests(TestCaseGui): def testOne(self): output = self.getOutput().split('\n') editor = self.getEditorText().split('\n') self.assertAlmostEqual(float(output[0]), math.atan2(1,1), 7, 'Checking first line') self.assertAlmostEqual(float(output[1]), math.atan2(1,-1), 7, 'Checking second line') self.assertAlmostEqual(float(output[2]), math.atan2(-1,1), 7, 'Checking third line') self.assertAlmostEqual(float(output[3]), math.atan2(-1,-1), 7, 'Checking fourth line') self.assertFalse(re.search(r'(0.785398|2.3561)', '\n'.join(editor)), 'Checking for hardcoding') self.assertFalse(re.search(r'( *import *math *as| *from *math)', '\n'.join(editor)), 'Checking import statement') myTests().main()

Use the double angle formula,

\[\sin(2\theta) = 2\sin(\theta)\cos(\theta),\]

to compute and print

\[\sin(\pi/4)\]

==== from unittest.gui import TestCaseGui import re class myTests(TestCaseGui): def testOne(self): output = self.getOutput() editor = self.getEditorText() self.assertAlmostEqual(float(output), 2*math.sin(math.pi/8)*math.cos(math.pi/8), 7, 'Checking output') self.assertFalse(re.search(r'(0.70710|math.sin *\( *math.pi */ *4 *\))', editor), 'Checking for hardcoding') self.assertFalse(re.search(r'( *import *math *as| *from *math)', editor), 'Checking import statement') myTests().main()

Q-1: Which is the best way to import the math module so you can use the exp() function?

• import math
• Correct! This will make it obvious to anyone who reads your code when you use something from the ``math`` module.
• from math import exp
• This imports only the ``exp()`` function and puts it in the global namespace. This can make it difficult to tell where this function comes from.
• from math import *
• This imports everything from ``math`` and puts it in the global namespace. This can make it difficult to tell when the ``math`` module is being used.
• import math as m
• Don't change the names of modules unless it is the convention for that module. This makes your code harder to read.

Q-1: Which of the following functions are part of the math module?

• exp()

• abs()

• int()

• sin()

• fabs()

• pi

Write a program to convert square km to square meters.

====

In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that

\[\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - ...\]

Write a program to calculate π when n is 99 and 999. When n becomes bigger, does π becomes more accurate?

====

Write a program to print an n*n times table. n could be any number. For example, the following is a 5*5 times table:

1 x 1 = 1

1 x 2 = 2 2 x 2 = 4

1 x 3 = 3 2 x 3 = 6 3 x 3 = 9

1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 4 x 4 = 16

1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20 5 x 5 = 25

====

Write a Python program to print 10 random integers between 1 and 100, inclusive. Print one random integer per line. Use a loop. I have started the program for you.

# replace this with a comment containing your name import random # your code below ====

Q-1: (TEST) The correct code to generate a random number between 1 and 100 (inclusive) is:

• prob = random.randrange(1, 101)
• This will generate a number between 1 and 101, but does not include 101.
• prob = random.randrange(1, 100)
• This will generate a number between 1 and 100, but does not include 100. The highest value generated will be 99.
• prob = random.randrange(0, 101)
• This will generate a number between 0 and 100. The lowest value generated is 0. The highest value generated will be 100.
• prob = random.randrange(0, 100)
• This will generate a number between 0 and 100, but does not include 100. The lowest value generated is 0 and the highest value generated will be 99.

Q-1: Which of the following is NOT deterministic?

• A program that reads in a data file of historical rainfall statistics, and predicts future rainfall as the average of the past 10 years' rainfall statistics.
• "Deterministic" means if the program is run with the same inputs, it will always produce the same output.
• A program that asks the user to think of a number from 1 - 100. The program tries to find the user's number through bisection search, repeatedly outputting a guess and asking the user to input whether the guess is correct, too high, or too low.
• "Deterministic" means if the program is run with the same inputs, it will always produce the same output.
• A program that picks a secret number randomly, then asks the user to guess the number. At each step, the computer outputs whether the guess is correct, too high, or too low.
• Since this program includes a random number being generated, the results will not always be the same, even given the same inputs.
• A program that asks the user to input a number, then computes the square root of the number using the Newton-Raphson method.
• "Deterministic" means if the program is run with the same inputs, it will always produce the same output.

Q-1: What is the approximate probability of rolling a 3 at least once in 5 die rolls?

• 0.5
• It is 1 minus the probability of getting no 3's, so 1 - (5/6)^5.
• 0.6
• It is 1 minus the probability of getting no 3's, so 1 - (5/6)^5.
• 0.7
• It is 1 minus the probability of getting no 3's, so 1 - (5/6)^5.
• 0.8
• It is 1 minus the probability of getting no 3's, so 1 - (5/6)^5.

Q-1: What is the approximate probability of rolling a die 5 times without getting a single 3?

• 0.2
• In a single die roll, the probability of NOT getting a 3 is 5/6. Each die roll is independent, so the probability of not getting a 3 in each of the 5 die rolls is (5/6)^5.
• 0.3
• In a single die roll, the probability of NOT getting a 3 is 5/6. Each die roll is independent, so the probability of not getting a 3 in each of the 5 die rolls is (5/6)^5.
• 0.4
• In a single die roll, the probability of NOT getting a 3 is 5/6. Each die roll is independent, so the probability of not getting a 3 in each of the 5 die rolls is (5/6)^5.
• 0.5
• In a single die roll, the probability of NOT getting a 3 is 5/6. Each die roll is independent, so the probability of not getting a 3 in each of the 5 die rolls is (5/6)^5.

Q-1: Given normally distributed data on a random variable of interest, approximately what percentage of the data falls within one standard deviation of the mean?

• 55%
• 68%
• 79%
• 95%

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