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181 lines
6.1 KiB
Python
181 lines
6.1 KiB
Python
import random
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import math
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def combinations(max_lengthgth=20):
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"""Combinations of Objects
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| Find the number of combinations from $19$ objects picked $6$ at a time. | $27132$ |
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"""
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a = random.randint(10, max_lengthgth)
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b = random.randint(0, 9)
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solution = int(
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math.factorial(a) / (math.factorial(b) * math.factorial(a - b)))
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problem = f"Find the number of combinations from ${a}$ objects picked ${b}$ at a time."
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return problem, f'${solution}$'
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def conditional_probability():
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"""Conditional Probability
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| Someone tested positive for a nasty disease which only $1.18$% of the population have. Test sensitivity (true positive) is equal to $SN=98.73$% whereas test specificity (true negative) $SP=99.99$%. What is the probability that this guy really has that disease? | $99.16$% |
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"""
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P_disease = round(2. * random.random(), 2)
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true_positive = round(random.random() + float(random.randint(90, 99)), 2)
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true_negative = round(random.random() + float(random.randint(90, 99)), 2)
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def BayesFormula(P_disease, true_positive, true_negative):
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P_notDisease = 100. - P_disease
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false_positive = 100. - true_negative
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P_plus = (P_disease) * (true_positive) + \
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(P_notDisease) * (false_positive)
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P_disease_plus = ((true_positive) * (100 * P_disease)) / P_plus
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return P_disease_plus
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answer = round(BayesFormula(P_disease, true_positive, true_negative), 2)
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problem = "Someone tested positive for a nasty disease which only ${0:.2f}$% of the population have. " \
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"Test sensitivity (true positive) is equal to $SN={1:.2f}$% whereas test specificity (true negative) $SP={2:.2f}$%. " \
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"What is the probability that this guy really has that disease?".format(
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P_disease, true_positive, true_negative)
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solution = f'${answer}$%'
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return problem, solution
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def confidence_interval():
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"""Confidence interval For sample S
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| The confidence interval for sample $[234, 223, 210, 203, 258, 299, 281, 208, 278, 252, 295, 245, 280, 235, 219, 297, 214, 267, 212, 256, 232, 221]$ with $99$% confidence is | $(263.31, 229.33)$ |
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"""
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n = random.randint(20, 40)
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j = random.randint(0, 3)
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lst = random.sample(range(200, 300), n)
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lst_per = [80, 90, 95, 99]
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lst_t = [1.282, 1.645, 1.960, 2.576]
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mean = 0
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sd = 0
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for i in lst:
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count = i + mean
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mean = count
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mean = mean / n
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for i in lst:
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x = (i - mean)**2 + sd
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sd = x
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sd = sd / n
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standard_error = lst_t[j] * math.sqrt(sd / n)
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upper = round(mean + standard_error, 2)
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lower = round(mean - standard_error, 2)
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problem = 'The confidence interval for sample ${}$ with ${}$% confidence is'.format(
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[x for x in lst], lst_per[j])
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solution = f'$({upper}, {lower})$'
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return problem, solution
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def data_summary(number_values=15, min_val=5, max_val=50):
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"""Mean, Standard Deviation and Variance
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| Find the mean,standard deviation and variance for the data $9, 29, 46, 27, 46, 15, 10, 44, 19, 33, 38, 7, 34, 28, 8$ | The Mean is $26.2$, Standard Deviation is $186.29$, Variance is $13.65$ |
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"""
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random_list = []
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for i in range(number_values):
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n = random.randint(min_val, max_val)
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random_list.append(n)
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a = sum(random_list)
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mean = round(a / number_values, 2)
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var = 0
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for i in range(number_values):
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var += (random_list[i] - mean)**2
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standardDeviation = round(var / number_values, 2)
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variance = round((var / number_values)**0.5, 2)
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problem = f"Find the mean,standard deviation and variance for the data ${', '.join(map(str, random_list))}$"
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solution = f"The Mean is ${mean}$, Standard Deviation is ${standardDeviation}$, Variance is ${variance}$"
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return problem, solution
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def dice_sum_probability(max_dice=3):
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r"""Probability of a certain sum appearing on faces of dice
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| If $2$ dice are rolled at the same time, the probability of getting a sum of $5 =$ | $\frac{4}{36}$ |
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"""
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a = random.randint(1, max_dice)
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b = random.randint(a, 6 * a)
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count = 0
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for i in [1, 2, 3, 4, 5, 6]:
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if a == 1:
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if i == b:
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count = count + 1
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elif a == 2:
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for j in [1, 2, 3, 4, 5, 6]:
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if i + j == b:
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count = count + 1
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elif a == 3:
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for j in [1, 2, 3, 4, 5, 6]:
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for k in [1, 2, 3, 4, 5, 6]:
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if i + j + k == b:
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count = count + 1
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problem = f"If ${a}$ dice are rolled at the same time, the probability of getting a sum of ${b} =$"
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solution = rf"\frac{{{count}}}{{{6**a}}}"
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return problem, solution
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def mean_median(max_length=10):
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"""Mean and Median
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| Given the series of numbers $[4, 19, 21, 22, 43, 44, 60, 81, 87, 92]$. Find the arithmatic mean and median of the series | Arithmetic mean of the series is $47.3$ and arithmetic median of this series is $43.5$ |
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"""
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randomlist = random.sample(range(1, 99), max_length)
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total = 0
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for n in randomlist:
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total = total + n
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mean = total / 10
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randomlist.sort()
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median = (randomlist[4] + randomlist[5]) / 2
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problem = f"Given the series of numbers ${randomlist}$. Find the arithmatic mean and median of the series"
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solution = f"Arithmetic mean of the series is ${mean}$ and arithmetic median of this series is ${median}$"
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return problem, solution
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def permutation(max_lengthgth=20):
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"""Permutations
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| Ex. Problem | Ex. Solution |
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| --- | --- |
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| Number of Permutations from $18$ objects picked $5$ at a time is: | $1028160$ |
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"""
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a = random.randint(10, max_lengthgth)
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b = random.randint(0, 9)
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solution = int(math.factorial(a) / (math.factorial(a - b)))
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problem = f"Number of Permutations from ${a}$ objects picked ${b}$ at a time is: "
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return problem, f"${solution}$"
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