hw06

hw06 November 30, 2023 [124]: # Initialize Otter import otter grader = otter . Notebook( "hw06.ipynb" ) 1 Homework 6: Probability, Simulation, Estimation, and Assess- ing Models Please complete this notebook by filling in the cells provided. Before you begin, execute the previous cell to load the provided tests. Helpful Resource: - Python Reference : Cheat sheet of helpful array & table methods used in Data 8! Recommended Readings : * Randomness * Sampling and Empirical Distributions * Testing Hypotheses Please complete this notebook by filling in the cells provided. Before you begin, execute the cell below to setup the notebook by importing some helpful libraries. Each time you start your server, you will need to execute this cell again. For all problems that you must write explanations and sentences for, you must provide your answer in the designated space. Moreover, throughout this homework and all future ones, please be sure to not re-assign variables throughout the notebook! For example, if you use max_temperature in your answer to one question, do not reassign it later on. Otherwise, you will fail tests that you thought you were passing previously! Deadline: This assignment is due Wednesday, 10/4 at 11:00pm PT . Turn it in by Tuesday, 10/3 at 11:00pm PT for 5 extra credit points. Late work will not be accepted as per the policies page. Note: This homework has hidden tests on it. That means even though tests may say 100% passed, it doesn’t mean your final grade will be 100%. We will be running more tests for correctness once everyone turns in the homework. Directly sharing answers is not okay, but discussing problems with the course staff or with other students is encouraged. Refer to the policies page to learn more about how to learn cooperatively. You should start early so that you have time to get help if you’re stuck. Offce hours are held Monday through Friday in Warren Hall 101B. The offce hours schedule appears here . 1

1.1 1. Roulette [125]: # Run this cell to set up the notebook, but please don't change it. # These lines import the Numpy and Datascience modules. import numpy as np from datascience import * # These lines do some fancy plotting magic. import matplotlib % matplotlib inline import matplotlib.pyplot as plt plt . style . use( 'fivethirtyeight' ) import warnings warnings . simplefilter( 'ignore' , FutureWarning ) A Nevada roulette wheel has 38 pockets and a small ball that rests on the wheel. When the wheel is spun, the ball comes to rest in one of the 38 pockets. That pocket is declared the winner. The pockets are labeled 0, 00, 1, 2, 3, 4, … , 36. Pockets 0 and 00 are green, and the other pockets are alternately red and black. The table wheel is a representation of a Nevada roulette wheel. Note that both columns consist of strings. Below is an example of a roulette wheel! Run the cell below to load the wheel table. [126]: wheel = Table . read_table( 'roulette_wheel.csv' , dtype = str ) wheel [126]: Pocket | Color 00 | green 0 | green 1 | red 2 | black 3 | red 4 | black 5 | red 6 | black 7 | red 8 | black … (28 rows omitted) 1.1.1 Betting on Red If you bet on red , you are betting that the winning pocket will be red. This bet pays 1 to 1 . That means if you place a one-dollar bet on red, then: • If the winning pocket is red, you gain 1 dollar. That is, you get your original dollar back, plus one more dollar. • If the winning pocket is not red, you lose your dollar. In other words, you gain -1 dollars. 2

8 | black | -1 … (28 rows omitted) [131]: grader . check( "q1_2" ) [131]: q1_2 results: All test cases passed! 1.1.2 Simulating 10 Bets on Red Roulette wheels are set up so that each time they are spun, the winning pocket is equally likely to be any of the 38 pockets regardless of the results of all other spins. Let’s see what would happen if we decided to bet one dollar on red each round. Question 3. Create a table ten_bets by sampling the table wheel to simulate 10 spins of the roulette wheel. Your table should have the same three column labels as in wheel . Once you’ve created that table, set sum_bets to your net gain in all 10 bets, assuming that you bet one dollar on red each time. (4 points) Hint: It may be helpful to print out ten_bets after you create it! [132]: ten_bets = wheel . sample( 10 ) sum_bets = np . sum(ten_bets . column( 'Winnings: Red' )) sum_bets [132]: 0 [133]: grader . check( "q1_3" ) [133]: q1_3 results: All test cases passed! Run the cells above a few times to see how much money you would make if you made 10 one-dollar bets on red. Making a negative amount of money doesn’t feel good, but it is a reality in gambling. Casinos are a business, and they make money when gamblers lose. Question 4. Let’s see what would happen if you made more bets. Define a function net_gain_red that takes the number of bets and returns the net gain in that number of one-dollar bets on red. (4 points) Hint: You should use your wheel table within your function. [134]: def net_gain_red (n): n_bets = wheel . sample(n) return (np . sum(n_bets . column( 'Winnings: Red' ))) [135]: grader . check( "q1_4" ) [135]: q1_4 results: All test cases passed! Run the cell below a few times to make sure that the results are similar to those you observed in the previous exercise. 4

[136]: net_gain_red( 10 ) [136]: 0 Question 5. Complete the cell below to simulate the net gain in 200 one-dollar bets on red, repeating the process 10,000 times. After the cell is run, simulated_gains_red should be an array with 10,000 entries, each of which is the net gain in 200 one-dollar bets on red. (4 points) Hint: Think about which computational tool might be helpful for simulating a process multiple times. Lab 5 might be a good resource to look at! Note: This cell might take a few seconds to run. [137]: num_bets = 200 repetitions = 10000 simulated_gains_red = [] for i in range (repetitions): simulated_gains_red . append(net_gain_red(num_bets)) len (simulated_gains_red) # Do not change this line! Check that ␣ ↪ simulated_gains_red is length 10000. [137]: 10000 Run the cell below to visualize the results of your simulation. [138]: gains = Table() . with_columns( 'Net Gain on Red' , simulated_gains_red) gains . hist(bins = np . arange( -80 , 41 , 4 )) 5

[141]: def dollar_bet_on_split (n): if ((n == "5" ) or (n == "6" )): return 17 else : return -1 [142]: grader . check( "q1_7" ) [142]: q1_7 results: All test cases passed! Run the cell below to check that your function is doing what it should. [143]: print (dollar_bet_on_split( '5' )) print (dollar_bet_on_split( '6' )) print (dollar_bet_on_split( '00' )) print (dollar_bet_on_split( '23' )) 17 17 -1 -1 Question 8. Add a column Winnings: Split to the wheel table. For each pocket, the column should contain your gain in dollars if that pocket won and you had bet one dollar on the 5-6 split. (4 points) [144]: split_winnings = wheel . apply(dollar_bet_on_split, "Pocket" ) wheel = wheel . with_column( "Winnings: Split" ,split_winnings) wheel . show( 5 ) # Do not change this line. <IPython.core.display.HTML object> [145]: grader . check( "q1_8" ) [145]: q1_8 results: All test cases passed! Question 9. Simulate the net gain in 200 one-dollar bets on the 5-6 split, repeating the process 10,000 times and saving your gains in the array simulated_gains_split . (5 points) Hint: Your code in Questions 4 and 5 may be helpful here! [146]: simulated_gains_split = [] for i in range ( 10000 ): net = np . sum(wheel . sample( 200 ) . column( "Winnings: Split" )) simulated_gains_split . append(net) # Do not change the two lines below gains = gains . with_columns( 'Net Gain on Split' , simulated_gains_split) gains . hist(bins = np . arange( -200 , 150 , 20 )) 7

Question 10. Look carefully at the visualization above, and assign histogram_statements to an array of the numbers of each statement below that can be correctly inferred from the overlaid histogram. (2 points) 1. If you bet one dollar 200 times on a split, your chance of losing money is more than 50%. 2. If you bet one dollar 200 times in roulette, your chance of making more than 50 dollars is greater if you bet on a split each time than if you bet on red each time. 3. If you bet one dollar 200 times in roulette, your chance of losing more than 50 dollars is greater if you bet on a split each time than if you bet on red each time. Hint: We’ve already seen one of these statements in a prior question. [147]: histogram_statements = np . array([ 1 , 2 , 3 ]) [148]: grader . check( "q1_10" ) [148]: q1_10 results: All test cases passed! If this exercise has put you off playing roulette, it has done its job. If you are still curious about other bets, here they all are, and here is the bad news. The house – that is, the casino – always has an edge over the gambler. 1.2 2. Chances Before you do this exercise, make sure you understand the logic behind all the examples in Section 9.5 . Good ways to approach probability calculations include: • Thinking one trial at a time: What does the first one have to be? Then what does the next one have to be? • Breaking up the event into distinct ways in which it can happen. • Seeing if it is easier to find the chance that the event does not happen. 8

• Event A: all four different animals are picked (assuming the child picks without replacement) • Event B: all four different animals are picked (assuming the child picks with replacement) [ ]: toys_option = 1 [ ]: grader . check( "q2_5" ) Question 6. In a lottery, two numbers are drawn at random with replacement from the integers 1 through 1000. (2 points) • Event A: The number 8 is picked on both draws • Event B: The same number is picked on both draws [ ]: lottery_option = 3 [ ]: grader . check( "q2_6" ) Question 7. A fair coin is tossed repeatedly. (2 points) • Event A: There are 60 or more heads in 100 tosses • Event B: There are 600 or more heads in 1000 tosses Hint : Think about the law of averages! [ ]: coin_option = 1 [ ]: grader . check( "q2_7" ) 1.3 3. Three Ways Python Draws Random Samples You have learned three ways to draw random samples using Python: • tbl.sample draws a random sample of rows from the table tbl . The output is a table consisting of the sampled rows. • np.random.choice draws a random sample from a population whose elements are in an array. The output is an array consisting of the sampled elements. • sample_proportions draws from a categorical distribution whose proportions are in an array. The output is an array consisting of the sampled proportions in all the categories. [ ]: # Look through this code and run this cell for questions 1 and 2 top_movies = Table . read_table( 'top_movies_2017.csv' ) . select( 0 , 1 ) top_movies . show( 3 ) [ ]: # Look through this code and run this cell for questions 1 and 2 studios_with_counts = top_movies . group( 'Studio' ) . sort( 'count' , descending = True ) studios_with_counts . show( 3 ) [ ]: # Look through this code and run this cell for questions 1 and 2 studios_of_all_movies = top_movies . column( 'Studio' ) 10

distinct_studios = studios_with_counts . column( 'Studio' ) print ( "studios_of_all_movies:" , studios_of_all_movies[: 10 ], "..." ) print ( " \n distinct_studios:" , distinct_studios) [ ]: # Look through this code and run this cell for questions 1 and 2 studio_counts_only = studios_with_counts . column( 'count' ) studio_proportions_only = studio_counts_only / sum (studio_counts_only) print ( "studio_counts_only:" , studio_counts_only) print ( " \n studio_proportions_only:" , studio_proportions_only) [ ]: top_movies . sample( 10 ) [ ]: np . random . choice(studios_of_all_movies, 10 ) [ ]: sample_proportions( 10 , studio_proportions_only) [ ]: studio_proportions_only In Questions 1 and 2, we will present a scenario. For each scenario, we will ask whether the desired result can be achieved by using a given function and the following tables/arrays: top_movies , studios_with_counts , studios_of_all_movies , distinct_studios , studio_counts_only and studio_proportions_only . Note: Do not explain your answer; please answer yes or no and the name of the array/table. Question 1. Simulate a sample of 10 movies drawn at random with replacement from the 200 movies. Using just this sample, do we have enough information to output True if Paramount appears more often than Warner Brothers among studios that released the sampled movies, and False otherwise? Yes, with “top_movies” Note : Do not explain your answer for any of the options you’ve chosen; please follow the structure of the example answer provided. Question 1(a) Can this be done using the sample function? If yes, what table would we call sample on? (1 point) Yes, on “top_movies” Question 1(b) Can this be done using the np.random.choice function? If yes, what array would we call np.random.choice on? (1 point) Yes, on “studios_of_all_movies” Question 1(c) Can this be done using the sample_proportions function? If yes, what array would we call sample_proportions on? (1 point) Yes, on “studio_proportions_only” 11

[151]: grader . check( "q4_1" ) [151]: q4_1 results: All test cases passed! Question 2. We believe Jade’s model is incorrect. In particular, we believe there to be a larger chance of getting a face card. Which of the following statistics can we use during our simulation to test between the model and our alternative? Assign statistic_choice to the correct answer. (3 Points) 1. The distance (absolute value) between the actual number of face cards in 13 draws and 4, the expected number of face cards in 13 draws 2. The expected number of face cards in 13 draws 3. The number of face cards we get in 13 draws [152]: statistic_choice = 1 statistic_choice [152]: 1 [153]: grader . check( "q4_2" ) [153]: q4_2 results: All test cases passed! Question 3. Define the function deck_simulation_and_statistic , which, given a sample size and an array of model proportions (like the one you created in Question 1), returns the number of face cards in one simulation of drawing cards under the model specified in model_proportions . (5 Points) Hint: Think about how you can use the function sample_proportions . [154]: def deck_simulation_and_statistic (sample_size, model_proportions): return ((sample_proportions(sample_size, model_proportions) * sample_size) . ↪ item( 0 )) deck_simulation_and_statistic( 13 , deck_model_probabilities) [154]: 7.0 [155]: grader . check( "q4_3" ) [155]: q4_3 results: All test cases passed! Question 4. Use your function from above to simulate the drawing of 13 cards 5000 times under the proportions that you specified in Question 1. Keep track of all of your statistics in deck_statistics . (5 Points) [156]: repetitions = 5000 deck_statistics = [] for i in range (repetitions): 13

deck_statistics . ↪ append(deck_simulation_and_statistic( 13 ,deck_model_probabilities)) deck_statistics [156]: [6.0, 1.0, 4.0, 5.0, 5.0, 5.0, 6.0, 5.0, 4.0, 4.0, 5.0, 6.0, 5.0, 2.0, 5.0, 4.0, 3.0, 5.0, 3.0, 3.0, 5.0, 1.0, 2.0, 2.0, 5.0, 7.0, 2.0, 5.0, 5.0, 3.0, 4.0, 3.0, 2.0, 6.0, 3.0, 6.0, 2.0, 5.0, 2.0, 2.0, 6.0, 3.0, 14

5.0, 4.0, 2.0, 5.0, 4.0, 2.0, 4.0, 5.0, 3.0, 1.0, 6.0, 2.0, 2.0, 1.0, 5.0, 2.0, 5.0, 6.0, 4.0, 3.0, 5.0, 3.0, 5.0, 4.0, 5.0, 4.0, 4.0, 4.0, 3.0, 6.0, 5.0, 6.0, 6.0, 6.0, 6.0, 4.0, 6.0, 6.0, 4.0, 5.0, 4.0, 4.0, 5.0, 3.0, 3.0, 2.0, 8.0, 16

2.0, 5.0, 1.0, 5.0, 5.0, 4.0, 5.0, 5.0, 3.0, 3.0, 4.0, 3.0, 4.0, 4.0, 6.0, 4.0, 6.0, 5.0, 3.0, 3.0, 4.0, 5.0, 4.0, 6.0, 5.0, 6.0, 5.0, 5.0, 3.0, 6.0, 5.0, 3.0, 3.0, 4.0, 7.0, 3.0, 4.0, 2.0, 3.0, 9.0, 5.0, 5.0, 5.0, 4.0, 4.0, 5.0, 3.0, 17

1.0, 4.0, 1.0, 6.0, 6.0, 2.0, 7.0, 3.0, 6.0, 5.0, 3.0, 3.0, 7.0, 5.0, 5.0, 6.0, 6.0, 6.0, 4.0, 3.0, 3.0, 4.0, 4.0, 4.0, 5.0, 5.0, 2.0, 1.0, 5.0, 4.0, 3.0, 4.0, 6.0, 3.0, 3.0, 3.0, 3.0, 3.0, 4.0, 1.0, 2.0, 5.0, 3.0, 4.0, 2.0, 3.0, 2.0, 19

3.0, 2.0, 4.0, 4.0, 2.0, 4.0, 6.0, 0.0, 3.0, 1.0, 6.0, 3.0, 4.0, 7.0, 4.0, 4.0, 3.0, 7.0, 7.0, 3.0, 5.0, 5.0, 4.0, 6.0, 3.0, 3.0, 4.0, 2.0, 7.0, 4.0, 8.0, 2.0, 2.0, 5.0, 5.0, 5.0, 4.0, 4.0, 5.0, 3.0, 3.0, 4.0, 3.0, 5.0, 4.0, 4.0, 3.0, 20