4.3a Simulating Experiments Target Goal: I can use simulation to represent an experiment. In class FR

Question: In a class of 23 unrelated students (no twins!), what is the chance that at least two students have the same birthday? In a class of 23 unrelated students (no twins!), what is the chance that at least two students have the same birthday? In a room of 41 people? In a room of 41 people? In a room of 50 people? In a room of 50 people?

To determine the chance of this event, we can: 1.Conduct an experiment and measure the outcomes. Problem: we have to find many classes of the correct size; a lot of time & effort. Problem: we have to find many classes of the correct size; a lot of time & effort. REPLICATION

2.Calculate the theoretical probability using the mathematical laws of probability. Problem: the formulas can be complicated or involve higher mathematics. Problem: the formulas can be complicated or involve higher mathematics.

Simulation 3.Simulate an experiment using a model that is similar to this real life event. An that uses event is called An that uses event is called simulation. A well-designed model will yield accurate results for a large number of trials. imitation of chance behavior a model to imitate a real life

Independence Is Important! The result of one toss of a coin, for example, has no effect on the next toss.

Steps Involved in Simulation 1.State the problem or describe the experiment. 2.State the assumptions. 3.Assign a digit to each outcome. 4.Simulate many repetitions. 5.State your conclusions.

Example: Birthday 1.In a class of 23 unrelated students, what is the chance that at least two students have the same birthday? 2.We assume all possible birthdays are equally likely, and that one student’s birthday is independent of the other students’ birthdays.

3.Assign numbers to each day of the year. 4.Randomly choose 23 numbers and look to see if any are duplicated. Record the results in a table. randInt (1, 365, 23)  L1 Repeat until you have completed many (10) trials. Fill in following table. (20 or more would be better. Do 10 for time sake.)

Sort data and repeat for 10 trials. Record in table.

5. In a class of 23 unrelated students, the chance that at least two students have the same birthday is approximately _______. Number of students with the same birthday Tally Relative Frequency None At least 2 Total # of Trials

Example 5.21 A Girl in the Family A couple wants a girl or 4 children, whichever comes first. What are the chances they will have a girl among their children? 1.Wants girl or 4 children at most. 2.Assume numbers in Table B equally likely and independent. 3.Table D: 0, 1, 2, 3, 4: girl 5, 6, 7, 8, 9: boy 5, 6, 7, 8, 9: boy

4.Read digits from Table D until couple has either a girl or 4 children. Use line 130 of Table B. (+ girl born, - no girl born): 25 trails 690/ BBG/BG + +

Our estimate of the probability that this strategy will produce a girl is in these 25 trails: Our estimate of the probability that this strategy will produce a girl is in these 25 trails: # that produced a girl = _____ = # trials

Ways to Assign Digits a) Table b) Calculator c) Computer software

Assigning Digits with Table B Example: Choose person at random from group which 70% are employed: Example: Choose person at random from group which 70% are employed: Table: one digit represents 1 person Label: 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 = not employed 7, 8, 9 = not employed

Example: Choose person at random from group which 73% employed: Example: Choose person at random from group which 73% employed: Table: two digits represent 1 person Label: 00, 01, 02, …, 72 = employed 73, 74, …, 99 = not employed 73, 74, …, 99 = not employed

“Randint” examples Randint (0,9,5): generates 5 random integers between 0 and 9. Randint (0,9,5): generates 5 random integers between 0 and 9. Could serve as a block of 5 vs. table. Randint (1,6,7): Randint (1,6,7): Could simulate rolling die 7 times.

Exercise (with shoulder partner) Establishing a Correspondence State how you would use the following aids to establish a correspondence in a simulation that involves a 75% chance. A coin: If you flip a coin twice, how many possible outcomes? 4 Let HH mean failure. Let HH mean failure. Let the other three outcomes Let the other three outcomes HT, TH, TT be success.

75% chance A six-sided die: Let 1, 2, 3 be success. Let 1, 2, 3 be success. Let 4 mean failure. Let 4 mean failure. If you roll a 5 or 6 ignore then and roll again.

75% chance A random digit table (Table B): Peel off two digit numbers from the table. Peel off two digit numbers from the table. Let 01 through 75 mean success. Let 01 through 75 mean success. Let 76 through 99 and 00 mean failure. Let 76 through 99 and 00 mean failure.

75% chance A standard deck of playing cards: Let hearts, diamonds, and spade be success. Let club be failure.