Simulations PROBABILITY RECAP What is a probability The

Simulations

PROBABILITY RECAP

What is a probability? • The probability of an event refers to the likelihood that the event will occur. • EXAMPLE. The probability of winning your sports game this weekend is 0. 7 • What does that mean

Remember •

Convert these probabilities FRACTION (simplify) DECIMAL PERCENTAGE 4/10 0. 6 10% 1/5 0. 75

FRACTION 4/10 DECIMAL PERCENTAGE 0. 4 3/5 1/10 0. 6 40% 60% 1/5 3/4 0. 1 0. 2 0. 75 10% 20% 75%

What are the probabilities of these events? • • • Getting a head when you flip a coin? Waking up on a day of the week that begins with T Waking up on a weekday Waking up on a day where you need to go to school Rolling an even number on a dice Rolling an odd number on a dice

What is a simulation • Simulation is a way to model random events, such that simulated outcomes closely match real -world outcomes. • By observing simulated outcomes, researchers gain insight on the real world.

Why use simulation? • Some situations do not lend themselves to precise mathematical treatment. • Others may be difficult, time-consuming, or expensive to analyse. • In these situations, simulation may approximate real-world results; yet, require less time, effort, and/or money than other approaches.

Remember • A simulation is useful only if it closely mirrors real-world outcomes.

Example On average, Freddy sinks a 3 pointer in basketball once in every 10 shots, and suppose he gets exactly two opportunities to shoot in every game. Using simulation, estimate the likelihood that she will land two three pointer in a single game.

Simulations This is an experiment in which the conditions of a real life situation is reproduced We need to use a random number generator (calculator, dice, cards etc) in order to carry out the experiment.

Notes: How to design a simulation TTRC TOOL: How will you generate random numbers? What does the digit represents? Decimal points? TRIALS: Trial consist of? A successful trial? How many trials? (should always do at least 30) RESULTS: TABLE CALCULATION: Calculate probability or the mean to answer the question

How to use the “tool” How can we use a pack of cards to represent a die? Sally goes to the bathroom 4 times during a 6 hour work period KFC gives out 7 toy figurines 10% of all batteries are faulty

Describe the best tool to represent… 1. 75% of students pass Maths 2. 10% of buses are late 3. 1 out of 10 people have hazel eyes, 3 have blue and the rest are brown 4. Half of students parents are still married 5. 1 out of 6 sheep give birth to triplets, 2 give birth to twins and the rest have singles. 6. Flipping two coins

Example to describing 1. Mr Peppers Dog “fluffy” will go toilet inside 2% of the time a day. Find how many times Fluffy will go toilet inside in a week?

Describe how you would model these situations 1. A battery factory distributes batteries in packs of 5. 5% of batteries are faulty. How many do you expect to be faulty in each pack? 2. The school bus is late 20% of the time. How many times will it be late in a 5 day week? 3. Coca-Cola has a cash reward going. You must collect all the letters (C. A. S. H) that appear under the cap to win. Each letter is equally likely. How many Coca-Cola’s will you have to buy to win

Notes: Example: Patrick is collecting a set of 3 different plastic toys from Mc. Donalds, which are available for 6 weeks. Patrick only visits Mc. Donalds once a week and will always receive a toy. The toys are distributed randomly and have the following probabilities TOY 1 2 3 0. 2 0. 5 Prob 1. Design a simulation to find the number of weeks Patrick will go to Mc. Donalds 2. Carry out the simulation 30 times. 3. What is the probability that Patrick will collect all toys within the 6 weeks? 4. Are there any assumptions you need to make?

TTRC Tool: I will generate random numbers between 1 and 10 on my calculator (10 RAN# +1), and I will ignore all decimals. The numbers will represent the following: 1, 2, 3 will represent toy 1 4, 5 will represent toy 2 6, 7, 8, 9, 10 will represent toy 3 Trial: I will generate 6 random numbers from 1 to 10 to represent the 6 weeks Mc. Donalds will have the toys available. The trial will finish after 6 weeks or when all 3 toys have been collected. A successful trial will be when all 3 toys are obtained. I will complete 30 trials.

TTRC Results: Trial Complete table 1 The average number 2 3 of visits to 4 Mc. Donalds is the 5 mean number of 6 weeks 7 8 Calculation: 9 Answer question… 10 11 12 13 14 15 Ran # Toy 1 Toy 2 Toy 3 1, 2, 3 4, 5 6, 7, 8, 9, 10 Weeks Y/N

Assumptions in TTRC? ? • The Toys are randomly distributed (customer does not get to choose) • All Toys are available at any one time • The probability of getting one toy does not effect the chance of getting another toy, i. e. they are independent • Patrick goes to Mc. Donalds once a week only • Patrick will get a toy every week

Standard assumptions (notes) • • Probability remains the same at all times Random distribution Availability Time frame

Conclusion You need to answer the question!! What is the number of visits require to collect all toys? ? 4. 7 visits What is the probability that he will collect all 3 toys? ? 67% WRITE IN CONTEXT AND SENSIBLE ROUNDING

Sampling Variability • Your simulation only produces an estimate of what is actually happening so. . If you did a another simulation you are likely to get a different estimate

Improvements • A better estimate would be to repeat the simulation several times and calculate the mean of all the estimates. • Increasing the number of trials – would give a better estimate of the mean number as any variations in results will have less impact on the overall estimate.

Potential Issues of Accuracy This depends on the distributing process. As if it is not randomly distributed then there will be bias. In terms of Mc. Donalds – it would depend on how the workers at Mc. Donalds distributed the toys. Example:

Things to think about. . • • • Colours/size/packaging? Favouritism? Unethical behaviour? Advertising? Other people decisions influence Anything else? ? How would these effect our simulation? ? The probabilities may change, or ….

State some assumptions, improvements and potential issues • Every time you go to the movies you collect a sticker. You need 4 stickers to get a free movie pass which lasts for 12 weeks. Assuming that Dan goes to the movies once a week, calculate the average amount of weeks Dan will go to the movies, and the probability he will get a free movie pass.

Exercises When Dingle Mouse is running his tail catches on fire 60% of the time. His ears catch on fire 20% of the time, and his whiskers 10% of the time. He never gets more than one thing on fire – that would be dangerous. Design a Simulation for Dingle Mouse and carry it out 30 times. 1. Use the results to estimate the probability that Dingle Mouse not catch fire?

Bad Jelly speeds on her broomstick 80% of the time to get to work. Mud Wiggle the worm sees 60% of the speeding offences and he writes a formal complaint letter. Bad Jelly knows that if Mud Wiggle has to write more than 2 letters she has successfully annoyed her. • Design and simulate this situation to find out how many times Bad Jelly needs to ride her broomstick to annoy Mud Wiggle. • Use your results to write a recommendation • What are the assumptions and limitations • What is the prob. that Bad Jelly only has to ride twice to annoy Mud Wiggle

Bad Jelly has 3 different animals that she can ride to work on a five-day week. Bad Jelly is happy when she gets to ride her Frog at least once a week. Camel Horse Frog 0. 1 0. 5 0. 4 • Design and describe a simulation • On average how many times will she get to ride her frog to work a week? • Are there any assumptions or limitations? • Find the prob. that bad jelly will be happy • Use theoretical prob. to show well your simulation works

When you play angry birds your chances of getting to the next level are: Getting there on first shot: 0. 8 Getting there on second shot: 0. 4 Getting there on third shot is 0. 3 Getting there on fourth shot is 0. 2

Robert has just lost his job and is worried about having enough money to feed his family. He considers the following option – stealing. If he gets caught he loses what he stole. Robert thinks this solution will work if he doesn’t get caught four times. Robbing a house Prob. of being successful 0. 4 Prob. of unsuccessful 0. 6 Prob. of getting caught 0. 5 1. Design and describe a simulation 2. Using your results write a recommendation about how many times Robert will have to rob a house to feed his family and whether this will work as a solution. 3. What are the assumptions of this simulations, how can you improve this? 4. What is the probability that Robert will only need to steal four times. 5. Use theoretical probability to show well your simulation worked

Problem As a part of Christmas advertising a petrol station gives away one of 6 Lego toys to each customer who purchases $20 or more of fuel. Calculate how many visits to the petrol station a customer would need to make on average to collect all 6 Lego toys. Assumption: The likelihood of one Lego toy being handed out is independent of another.

Solution (suggestion) Tool: Generate random numbers between 1 & 6 (inclusive), each number stands for each toy. Trial: One trial will consist of generating random numbers till all numbers from 1 to 6 have been generated. Count the number of random numbers need to get one full set Results: Trial Toy 1 Toy 2 Toy 3 Toy 4 Toy 5 Toy 6 1 Y Y Y 10 2 Y Y Y 19 Number of Trials needed: 30 would be sufficient Calculation: Average number of visits = Total visits Number of trials Tally Total Visits

Problem: Mary has not studied for her Biology test. She does not know any of the answers on a threequestion true-false test, and she decides to guess on all three questions Design a simulation to estimate the probability that Mary will ‘Pass’ the test. (i. e. guess correct answers to at least 2 of the 3 questions) Calculate theoretical probability that Mary will pass the test.

Solution (suggestion) Tool: The probability that Mary guesses a question true is one half. First digit using calculator 1 + 10 Ran# 1 to 5 stands for ‘correct answer’ 6 to 10 stands for ‘incorrect answer’ Trial: One trial will consist of generating 3 random numbers to simulate Mary answering one complete test. A successful outcome will be getting at least 2 of the 3 random numbers between 1 and 5. Results: Trial Outcome of Trial Result of Trial 1 122 Successful trial 2 167 Unsuccessful trial Number of Trials needed: 30 would be sufficient Calculation: Estimate of probability of ‘passing’ the exam =

Problem: Mary has not studied for her history test. She does not know any of the answers on an eight-question true-false test, and she decides to guess on all eight questions Design a simulation to estimate the probability that Mary will ‘Pass’ the test. (i. e. guess correct answers to at least 4 of the eight questions)

Solution (suggestion) Tool: The probability that Mary guesses a question true is one half. First digit using calculator 1 + 10 Ran# 1 to 5 stands for ‘correct answer’ 6 to 10 stands for ‘incorrect answer’ Trial: One trial will consist of generating 8 random numbers to simulate Mary answering one complete test. A successful outcome will be getting at least 4 of the 8 random numbers between 1 and 5. Results: Trial Outcome of Trial Result of Trial 1 12236754 Successful trial 2 13672987 Unsuccessful trial Number of Trials needed: 30 would be sufficient Calculation: Estimate of probability of ‘passing’ the exam =

Problem: Lotto 40 balls and to win you must select 6 in any order. In this mini Lotto, there are only 6 balls and you win when you select 2 numbers out of the 6. Design and run your own simulation to estimate the probability of winning (i. e. selecting 2 numbers out of the 6) Calculate theoretical probability of winning.

Solution (suggestion) Tool: Trial: Results: Two numbers (between 1 and 6) will need to be selected first (say 2 & 4) First digit using calculator 1 + 6 Ran#, ignore the decimals. One trial will consist of generating 2 random numbers Discard any repeat numbers A successful outcome will be getting 2 of the 6 random numbers generated Trial Outcome of Trial Result of Trial 1 2 4 Successful trial 2 13 Unsuccessful trial Number of Trials needed: 50 would be sufficient Calculation: Estimate of probability of ‘winning’ = Number of ‘successful’ outcome Number of trials Theoretical probability in this case is 1/15