CHAPTER 6 PROBABILITY & SIMULATION Section 6.1—SIMULATION

Section 6.1—Simulation Three methods of answering questions about chance occurrences: Estimate based on actual observations of the random phenomenon Develop a probability model and use it to calculate theoretical answers. Start with a model that reflects the truth about the random phenomenon and simulate many repetitions to estimate the chances.

Why is it good to use for finding the likelihood of certain results? What is simulation? It’s the imitation of chance behavior, based on a model that accurately reflects the experiment under consideration. Why is it good to use for finding the likelihood of certain results? Once we have a good model or a trustworthy way to simulate experiments, we can simulate many repetitions quickly. Also, because if we have more repetitions we’ll get a proportion of repetitions that will be close to the true population proportion. Simulation can give a pretty good estimate of the probabilities.

How do we simulate? We can use random digits from a table, graphing calculator, or computer software to simulate many repetitions quickly.

Simulation Steps Step 1: State the problem or describe the experiment. Step 2: State the assumptions. -There is no universal one, just basic assumptions you need to make for a specific scenario. Step 3: Assign digits to represent outcomes. Step 4: Simulate many repetitions. -(Remember estimated proportions become more accurate with more repetitions.) Step 5: Calculate relative frequencies and state your conclusions.

Find the Probability of winning the game A game of Chance Find the Probability of winning the game

The rules of the game A bag of marbles contains 9 blue and 1 green marble. If you pull the green marble from the bag you win the game. You are given 2 chances to pick a marble from the bag.

More Rules! If you pick the green marble on the first try you win and stop playing the game. If you pick a blue marble, 4 additional blue marbles are removed and you are given a second chance. If you pick a green marble on this try you win!

Determine the probability of winning Describe an appropriate random digit assignment and perform 20 simulations. Begin on line 127 on the random digit table. Determine the experimental probability of winning the game.

Step 1: State the problem or describe the experiment. There are 10 marbles: 9 blue, 1 green UP TO two chances to win. Choosing green wins. 1st chance: 1 green in 10 total marbles—10% chance 2nd chance: 1 green in 5 total marbles—20% chance What is the chance of winning this game?

Step 2: State the assumptions. We are assuming all of the marbles are the same size and therefore have an equal chance of being selected on any individual draw from the bag of marbles. Additionally, we are assuming that each selection from the bag in a given round is independent from another. Essentially, if I were to repeat the first round over and over again, my first selection would not impact my second selection.

Step 3: Assign digits to represent outcomes. We need two unique assignments: one for each round. In each case, a single digit represents a pull of a marble from the bag. 1st Attempt: 0green, win, game over 1-9blue, move on to second attempt 2nd Attempt: 0-1green, win, game over 2-9blue, lose, game over

Step 4: Simulate many repetitions. Looking at one or two digits at a time will simulate one “play” of this game. If the first digit is a 0 then the game is over and only one digit is needed. If the first digit is not a 0 then a second digit is needed. So, we will need at least 20 digits and at most 40 digits to simulate playing the game 20 times. 4-3-L, 9-0-W, 9-9-L, 9-4-L, 7-7-L, 2-5-L, 3-3-L, 0-W, 6-4-L, 3-5-L, 9-4-L, 0-W, 0-W, 8-5-L, 1-6-L, 9-2-L, 5-8-L, 5-1-W, 1-7-L, 3-6-L

Step 5: Calculate relative frequencies and state your conclusions. So, as we can see from our 20 simulations, we would win 5 out of 20 times. We have an experimental probability of 5/20 or 0.25. In other words, based on our simulation, we would expect someone to one this game roughly 25% of the time they would play. Do be more confident in our estimate, we would want to simulate many more repetitions (known as trials) which is easier to do with a computer. We could establish an estimate closer to the theoretical probability of winning which is 0.28.

Using the Calculator If you want to perform the same simulation with the same digit assignments then type in the following: randInt(0,9,100)L1:randInt(0,9,100)L2:((L1=0) or (L2=0) or (L2=1))L3:sum(L3)/100 When you do this, it is essentially repeating the simulation 100 times. You can change the 100 in all three spots to change the number of simulations. The number that is reported is the proportion of times that the game is won.