Welcome to CS 2

• Lecture will start at 10:40.• As usual, lecture will be recorded.• If you have any question, please ask in the chat.

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5: RandomNumbers and Simulations

Today• Random numbers‣What? How?‣Python module: random

• Computer simulations• Homework 3‣Dice simulation‣Tournament simulation

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Random Number• Random number‣ A number “chosen” randomly (in a set of values)‣ Example: Throwing a dice random number in {1,…,6}

• Random number generator (RNG)‣ “Device that generates a sequence of numbers or symbols that

cannot be reasonably predicted …”• Pseudorandom number generator (PRNG)‣ Algorithm that deterministically generates random numbers!‣ Contradiction?

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https://en.wikipedia.org/wiki/Random_number_generationhttps://en.wikipedia.org/wiki/Pseudorandom_number_generator

PRNG Idea• Use “complex” formulas to generate sequence

of numbers that looks random• Example: LCG (Linear congruential generator)

𝑋𝑋𝑛𝑛+1 = 𝑎𝑎 ∗ 𝑋𝑋𝑛𝑛 + 𝑏𝑏 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑋𝑋0 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖 𝑣𝑣𝑎𝑎𝑖𝑖𝑣𝑣𝑣𝑣

with large 𝑎𝑎, 𝑏𝑏, 𝑚𝑚.

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Why Random Number• Applications of random numbers‣ Simulations of non-deterministic events‣ Cryptography: generating keys‣ Games: generating levels, AI players, …‣ …

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Random Number in Python• random module (doc)• random.random() function‣ Generates random floating number in [0, 1)

‣ Can also be used to generate “other random numbers” y is a random number in [5, 15)

d is a random value in {1, …, 6}7

import random

x = random.random() # returns a random number# between 0 and 0.9999…

y = 10 * random.random() + 5

d = int(6 * random.random()) + 1

https://docs.python.org/3/library/random.html

Random Number in Python• Many (useful) functions in random module‣ random.randint(a,b) random integer N with a ≤ N ≤ b‣ random.choice(L) random element from the list L‣ random.gauss(mu, sigma) sample of a Gaussian distribution‣ random.shuffle(L) shuffle the list L ‣ …

• Initialization of the PRNG‣ random.seed(n) Initialize the generator with the given seed‣ Extremely useful to reproduce experiments!

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ComputerSimulations

Computer Simulation• Computer simulations (wiki)‣Useful to analyze complex systems‣May replace/confirm costly experiments‣May be used for prediction (e.g. weather forecast)‣…

• Often based on random simulations‣Simple example: Throwing a die

• Model-based simulations‣bad model bad simulations incorrect conclusions!10

https://en.wikipedia.org/wiki/Computer_simulation

Dice Simulation

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import random # Module needed to generate "random" numbersrandom.seed(123)

def one_dice(nb_simulations):counter = [0,0,0,0,0,0] # shorter version: counter = [0]*6

for i in range(nb_simulations):x = random.randint(1,6) # Throw a dice and get a random (uniform) resultcounter[x-1] += 1 # Be careful x is in {1,...,6}, but lists are indexed from 0

print("Raw values:", counter)

for i in range(6):# Compute percentage and round with 2 digits after the decimal pointcounter[i] = round( counter[i] * 100 / nb_simulations, 2)

print("Percentage values:", counter) # slightly different in Google Colab Notebook

# Run the simulationone_dice(10000)

Homework 3 – Part 1• Write a new function two_dice(nb_sims) that simulates

throwing two dice and compute the sum of the dice values.• Print the percentage of time each sum is obtained (sum can

be 2, 3, 4, …, 12).

• Submit on OCWi before next lecture (Jan 25, 10:40).‣ Submit only two_dice function (I do not need the whole notebook)‣ 5 points

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TournamentSimulation

Tournament Simulation• Motivation‣Let's simulate a tournament and estimate the winning chance of each player‣Not always the best players in top positions

• Questions‣ Is it normal/usual?‣ Is it possible to be lucky (and win while being weaker)? Let's try to find out with computer simulations!

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Tournament Model• Tournament‣32 players‣Single-elimination bracket (loss eliminated) (wiki)‣ Initial random allocation of players‣Third place playoff for semifinals losers (wiki) Four top players win points; 100, 60, 30, 10 points

• Players‣Random strength in {0,…,100}‣Stronger player always wins against weaker players

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Adrian

Kevin

Lisa

Scott

Bert

Zaphod

Drew

Ernie

Michelle

Andrew

Johan

Ian

Monica

Steve

Red

Robert

Kevin

Lisa

Zaphod

Ernie

Lisa

Ernie

Andrew

Ian

Monica

Robert

Andrew

Robert

https://en.wikipedia.org/wiki/Single-elimination_tournamenthttps://en.wikipedia.org/wiki/Third_place_playoff

Player Generation

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# generate_players_strength.py# I use this file to generate 32 players

import randomrandom.seed(2021) # Fix the PRNG seed for reproducibility

NB_PLAYERS = 32 # Number of playersSTRENGTH = [] # Array (list) to store the strength of each player

# Generate random strength for each playerfor _ in range(NB_PLAYERS): # _ can be used when the index is not used

s = random.randint(0, 100) # obtain a random strength between 0 and 100 (incl.)STRENGTH.append(s)

print(STRENGTH)# [51, 80, 69, 35, 31, 81, 4, 56, 60, 73, 8, 40, 34, 37, 60, 9, 20, 21, 6, 95, 13, 86, 91, 70, 37, 6, 68, 87, 59, 14, 25, 77]

STRENGTH = [random.randint(0,100) for _ in range(NB_PLAYERS)] # shorter version# This is called list comprehension; specific to Python!

Tournament

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# tournament.py# Global variablesNB_PLAYERS = 32STRENGTH = [51, 80, 69, 35, 31, 81, 4, 56, 60, 73, 8, 40, 34, 37, 60, 9, 20, 21, 6, 95, 13, 86, 91, 70, 37, 6, 68, 87, 59, 14, 25, 77]

def compute_winner(player1, player2):""" Compute the winner in a match between two players."""

def simulate_tournament():""" Simulate a tournament with all NB_PLAYERS players1) Generate a random bracket2) Simulate the competition3) Return the identities of the players ranked 1 to 4"""

def compute_stats(nb_simu):""" Compute statistics based on nb_simu tournaments."""

Homework 3 – Part 2• Tournament model‣Stronger player always wins against weaker players‣Unrealistic assumption Let's try to improve the model

• Non-deterministic outcome of a game:‣Player1 with strength S1 versus Player2 with strength S2:

P1 wins with proba. 𝑆𝑆1𝑆𝑆1+𝑆𝑆𝑆

and P2 wins with proba. 𝑆𝑆𝑆𝑆𝑆1+𝑆𝑆𝑆

‣Example: Player of strength 30 wins 1/3 of the games vs a player of strength 60

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Homework 3 – Part 2• Write a new function compute_winner(p1, p2) that

computes a winner according to the probabilities given on previous slide.

• Using the same set of players' strength, simulate the average points earned by each player under this new model.

• Submit on OCWi before next lecture (Jan 25, 10:40).‣ Submit only compute_winner function‣ 10 points

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Advanced Remarks• Using only random.random(), how to implement

a function similar to:‣random.choice(L) (not too difficult)‣random.shuffle(L) (much more difficult)‣random.gauss(…) (math question, not CS question)

• Check the source code of random module

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import randomimport inspect

code = inspect.getsource(random.shuffle)print(code)

Welcome to CS 25: Random�Numbers and SimulationsTodayRandom NumberPRNG IdeaWhy Random NumberRandom Number in PythonRandom Number in PythonComputer�SimulationsComputer SimulationDice SimulationHomework 3 – Part 1Tournament�SimulationTournament SimulationTournament ModelPlayer GenerationTournamentHomework 3 – Part 2Homework 3 – Part 2Advanced Remarks