Applications of Number Theory in Computer Science Curriculum
Enriching Introductory Programming Courses with Non-Intuitive Probability Experiments Component Yana Kortsarts Computer Science Department, Widener University, Chester, PA Yulia Kempner Department of Computer Science, Holon Institute of Technology, Holon, Israel Introduction Probability theory is branch of mathematics that plays one of the central roles, not only in computer science, but also in science at large The focus of the current work is on the integration of the non-intuitive probability experiments into the
introductory programming course These examples do not require any knowledge of probability The value of the probability or the expectation is computed via a program that students run Goals and Objectives The proposed enrichment helps to achieve the following goals: Engaging students in experimental problem solving Increasing students motivation and interest in programming Increasing students attention and curiosity during the class Enhancing students programming skills
Non-Intuitive Testable Probabilities Non-intuitiveness of many probability statements Difficult for students to guess a correct answer Non-intuitiveness: Intuitive problem solving leads to the wrong answer Testability: The non-intuitive probability problem could be translated into a programming assignment. It is possible to compute, or closely approximate, the required probability or expectation by writing a program with a low running time The Averaging Principle While designing a numerical simulation it is important to
take into account that, given a single trial, the value reached can be far off the expectation We apply the averaging principle The program simulates the experiment many times; the results for all trials are added and then averaged at the end to obtain a final answer By simple laws of probability of sum of independent events , we know that the average value, produced by our program, will converge to the correct result Is your random number generator really random? It is known that randomized algorithms might perform poorly because the random number generator was faulty Checking the performance of the random number
generator provides a soft introduction to the subject of estimating random events An obvious remark is that we do not know how to find even a single random bit. Still, we do know how to create pseudo-random bits Is your random number generator really random? To evaluate the behavior of the random number generator, we ask students to consider the simple experiment of throwing a fair coin with H and T 1000000 times This is a single trial of the experiment, and we ask students to verify that, for example, the number of resulting Hs is about 500000. For this example, it is known that standard
deviation is n 500, n 1000000 4 Classical problem: "Let's Make a Deal" The Monty Hall Problem - is a probability puzzle loosely based on the American television game show Let's Make a Deal and named after the show's original host, Monty Hall There are three doors. Behind one of them there is an expensive car. Behind the two others there is, say, a goat Experiment is conducted as follows:
The player chooses a door The host opens the door since there are two doors with a goat, the host, who knows where the car is, is able, regardless of the choice of the player, to open a door that contains a goat Now, we have the chosen door, the open door, and the third one At that moment, the host asks if the player wants to switch to the other (closed) door The question is whether the player should switch Intuitive Answer
For almost all students, since the chosen door and the other closed door look the same, the intuitive answer is not to switch Explanation of the Correct Answer The correct answer: with probability 2/3, one should switch, and with probability 1/3, one should not switch. We can say by symmetry, that the car is behind door 1
If the player chooses 1, then he should not switch If he chooses 2, he should switch, and if he chooses 3, he should switch This gives probability 2/3 for changing; namely, in a large number of trials, it would be worth changing to the other closed door about 2/3 of the time Numerical Solution def makeDeal(n): doSwitch=0 noSwitch=0 for i in range(n): carDoor=random.randint(1,3) playerDoor=random.randint(1,3) if(carDoor==playerDoor): noSwitch+=1 else:
doSwitch+=1 return (float)(noSwitch)/n, (float)(doSwitch)/n Birthday Paradox The non-intuitive property: for 23 students in the class, there is a probability of slightly more than 50 percent that there are two students who have the same birthday (take in account only day and month) The intuition: it cannot be that among only 23 people, there are two that were born on the same date - 23 is so much smaller than 365 For 23 people in the class, the number of possible pairs is 253 = (23 22)/2. This now seems much closer to 365 For 30 students, the probability of two people being born on the same day is 0.7
The "strange " behavior of the classical algorithm to find the maximum Input: an array A[1..n] of distinct numbers. Output: M maximum value 1. M A 2. for i = 2 to n: if A[i] > M: M A[i] How many times will the maximum be exchanged if the order between numbers of the array is random? Typical guesses are n/2 or n/4. However, the expected number of exchanges is bounded by ln n + O(1) Theoretical Explanation Let X i be the indicator variable for the event that
M was replaced by A[i] Thus the number of exchanges is C X i i The expectation of C is E ( X i ) i The probability that A[i] will be the maximum of the first i elements, in a random array, is 1/i by symmetry, hence 1 E (C ) ln n O(1) i i Numerical Solution Since only the order of the numbers is important,
we recommend the use of numbers 1, 2, . . . , n as the entries of an array of size n To generate a random permutation of the numbers 1, 2, . . . , n, we recommend using the Fisher-Yates shuffle, also known as the Knuth shuffle with time complexity O(n) Numerical Solution Fisher-Yates (Knuth) Shuffle Initialization: for i from 1 to n do: A[i]=i Shuffle: for i from n to 2 do: j = random integer between 1 and i swap A[j] and A[i]
ln (1000) is about 6.9, students are able to observe the surprising result that maximum is changed only a few times The strange behavior of random walks In our example, the random walk starts at 0, and the goal is to reach n = 100 At 0, the random walk must move to 1. At each step, which is not 0, the random walk moves +1 or -1 with equal
probability 1/2 The process ends when the random walk reaches n = 100 Here, the students are asked to make predictions about the number of steps required to walk from 0 to 100 2 It turns out that the expectation is n Final Example This example illustrates the known problem of faults when a collection of bits is streamed Some 0 values change to 1 due to local interference and
vice versa The problem: A transmitter sends binary bits For a single bit, there is a probability 0.8 that 0 is sent and probability 0.2 that 1 is sent When a 0 is sent, a 0 will be received with probability 0.8 When a 1 is sent, the 1 will be received with probability 0.9 Question and Answer Question: If a 1 is received, what is the probability that a 1 was sent?
The correct value is 0.53 To provide an intuitive explanation of why the probability is so low, one may point out that 1 arrives not so frequently, since the probability that 1 is sent is only 0.2, and it makes a small sample To get good, results one needs to take a large sample This can be explained in relation to, for example, polls taken for elections Assessment Multiple lab sessions allowed comparison assessment Half of the lab sessions practiced repetition and decision structures on non-intuitive probability experiments, and the rest of the students practiced the same material on standard examples Combination of indirect and direct assessment tools
Level of engagement, interest, and curiosity during the course work, 1 very low and 5 very high. A total of 102 students from both institutions participated in the pre and post survey Table 1: Averages for each category Category Pre-Survey Probability Group Post-Survey Pre-Survey Probability Non Group Probability
Group Post-Survey Non Probability Group Engagement 3.7 4.1 3.8 3.7 Interest
4.3 4.5 4.1 4.2 Curiosity 3.9 4.5 4.0 3.9
Results Probability students showed increased engagement, interest, and curiosity after nonintuitive probability experiments were introduced during the laboratory sessions In addition to surveys, all students were administered the same exam to directly measure their understanding of the programming structures. The average grades for both groups of students, before and after the probability component in the next table Table 2: Average Grades Category Probability
Group Before Probability Component Probability Group After Probability Component Non Probability Group Before Non
Probability Group After Average Grade 78.9 80.2 78.85 78.7 While the average in the non-probability group remained about the same, the probability group performed slightly better on
the exam that was introduced after the probability component. This supports our additional claim that non-intuitive probability experiments have potential to improve student learning and comprehension of the repetition and decision programming structures. Summary and Future Plans In general, the assessment results support our opinion on advantages of the proposed methodology We are planning to continue inclusion of the probability experiments in the future iterations of the course To integrate the proposed ideas into advanced courses, some of the examples could be generalized
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