Chapter 5.3: Simulation

Random  We call a phenomenon RANDOM if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

Probability  The proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency.

5.3 Simulation What are the chances….we can try to estimate the likelihood of a result of interest by actually carrying out an experiment many times and calculating the relative frequency. An experiment might not be possible or be too expensive. …we can simulate the sitatuion

5.3 Simulation …that is we can start with a model that, in some fashion, reflects the truth about the experiment, and then develop a procedure of imitating ---or simulating - -- a number of repetitions of the experiment.

 The more repetitions, the closer a result’s occurrence will get to it’s true likelihood.  Independence: When the result of one trial (coin toss, dice roll) has no effect or influence on the next toss.

Example: Toss a coin 10 times, what is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? At your desk, toss the coin given to you 10 times. Record H or T for each toss. Did you have a run of three heads or three tails (yes or no)

If we use simulation, instead of actual practice, we can replicate this situation many times, very quickly, and get a more accurate likelihood, or probability. Let’s do it!

Simulation Steps  1. State the problem or describe the random phenomenon  2. State the Assumptions (there are 2)  3. Assign digits to represent outcomes (want efficiency)  4. Simulate many repetitions  5. State conclusion

Let’s practice step 3  Go to table B line 101  Write down 10 digits  Do you have 3 odds in a row? Do you have 3 evens in a row?

Be sure to keep track of whether or not the event we want (a run of at least 3 heads or at least 3 tails) occurs on each repetitions Here are the first 3 repetitions starting at line 101 in Table B. Digits: H/T: HHTTH HHTHT THHHT TTHHH HTTTH HTHHH Run of 3: YES YES YES

Now let’s run a simulation  Choose a line in Table B (different than your classmates).  On your worksheet (step 4), write down 4 groups of 10 numbers  Determine if you have a run of 3 heads or 3 tails for each group.  Let’s tally our results as a class, before we state a conclusion.

Assigning digits  Some ways more efficient than others.  Example: Choose a person at random from a group of which 70% are employed.  Example 2: Choose one person at random from a group of which 73% employed  Example 3: Choose one person at random from a group of which 50% are employed, 20% are unemployed, and 30% are not in the labor force:

Frozen Yogurt Sales example  Orders of frozen yogurt flavors (based on sales) have the following relative frequencies: 38% chocolate, 42% vanilla, 20% strawberry.  We want to simulate customers entering the store and ordering yogurt.  How would you simulate 10 frozen yogurt sales based on recent history using table?

Assigning digits to outcomes  A couple plans to have children until they have a girl or until they have 4 children, whichever comes first. What are the chances that they will have a girl among their children?

Randomizing with Calculator  Block of 5 random digits from table  Rolling a die 7 times  10 numbers from 00-99