Simulations

1. Simulations AP Statistics Presentation 3.10

2. Family Planning • A new couple decides that they would like to have 3 children? • The father would really like to have a daughter. • What is the probability that the couple will have at least one daughter among three children?

3. Family Planning • What is the probability that the couple will have at least one daughter among three children? • To answer this question, one might… • Conduct an observational study and start randomly selecting families with three children. They would then find the proportion of them that have at least one girl. That would cost some time and a lot of money (for the surveys, phone calls, etc.). • Design and run an experiment. Gather willing couples. Instruct them to have exactly three children, then measure the results to find the proportion. This might be unethical and take a lot of time. • Design and conduct a simulation. This takes very little time and money and can answer the question.

4. Family Planning • What is the probability that the couple will have at least one daughter among three children? • The probability of having a girl is essentially 50% or 0.5. • We could then simply use a coin to simulate having three children letting a head represent a girl and a tail represent a boy. • This would go much faster than the experiment. • So, we toss three coins at a time to simulate one family. This is one repetition. • Like sample size, we want lots of repetitions, so we might flip the coins a couple of hundred times to determine the probability of having at least one daughter.

5. Steps to a simulation #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions • The difference in these steps is that we will use random digits instead of coin flips.

6. Steps to a simulation • How the steps look in practice (with the family planning example).

7. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions What is the probability of a couple having at least one daughter given that they have three children? Steps to a Simulation

8. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions The probability of having a girl is 0.5 or 50%. The gender of children is independent (the notion that is the 1st child is a boy the chance of the 2nd being a boy is still 50%. Steps to a Simulation

9. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions We’ll use random digits for this one. A table of random digits consists of a long, long list of digits from 0 to 9. Since the probability is 50%, assign half the digits to represent a girl and the other half to represent a boy. The digit assignment is: 0-4 = Girl 5-9 = Boy Steps to a Simulation

10. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions First, define what one repetition is. One repetition is having three children. One repetition then consists of 3 random digits. Steps to a Simulation

11. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Consider the random digits: 8273140727806078240317300157649684178912 The first 3 digits are 827, so the first family had a boy, girl, boy. The second family had 314 or girl, girl, girl. The 3rd family had 072 or girl, boy, girl. Conduct this about 50 times and you’ve got a good estimate of the true chance. Steps to a Simulation

12. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Using your many repetitions, you have a good estimate of the true chance of having at least one girl in a family of 3 children. Steps to a Simulation

13. More Simulations • Let’s try steps #3 and #4 again, this time using the calculator.

14. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions We’ll use a calculator this time. The calculator has a random number generator built in. Since the probability is 50%, we just need the calculator to give us a random number of 1 or 2. The digit assignment is: 1 = Girl 2 = Boy The calculator can give us a 1 or 2 randomly by going to the math menu, then prb (probability), then rand (ti89) or randint (ti83/84). See the screenshots ti89 on the left, ti83/84 on the right. Steps to a Simulation On the 83/84s, you enter randint (low number, hi number, number of numbers). We want a number from 1 to 2 and we want 3 of them.

15. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Again, define what one repetition is. One repetition is having three children. One repetition then consists of 3 random digits. Steps to a Simulation

16. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions The ti89 needs to do one child at a time. The ti83/84 can generate as many at once as you like (three in our case to simulate three children) Steps to a Simulation

17. A Sample Problem to Try • Ichiro had a batting average of .350 in his rookie season. That is (not getting technical), he got a hit 35% of the time. • If he typically has 4 at bats (or chances to hit) each game, how often would you expect him to have a multi-hit game (2 or more hits in a game)?

18. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions If Ichiro (who gets a hit 35% of the time) typically has 4 at bats (or chances to hit) each game, how often would you expect him to have a multi-hit game (2 or more hits in a game)? Steps to a Simulation

19. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions The probability of getting a hit is 0.35 or 35%. Getting a hit is independent (the notion that you can get hot (streak) or cold (slump) is not considered. Steps to a Simulation

20. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Since the probability is 35%, assign 35 out of 100 possible numbers to a hit. Possible digit assignments: Random Digit Table 01-35 = Hit 36-00 = No Hit Calculator 1-35 = Hit 36-100 = No Hit Steps to a Simulation You could also reduce (like fractions) to assign digits if you like. Since 35/100=7/20 you could use: 1-7 = Hit 8-20 = No Hit

21. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions First, define what one repetition is. One repetition is four at-bats (or tries), which is one game. Let’s simulate 20 games. Steps to a Simulation

22. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Use RandInt(1,100,4) – see top screenshot In this 1st game, Ichiro got a hit in his 1st and 3rd tries (a multi-hit game). In his 2nd game, Ichiro only got 1 hit (in his 2nd try), no multi-hit game. Keep a tally as you repeat this (by just continually pressing enter). Multi-Hit Game: IIIIIIIII No Multi-Hit Game: IIIIIIIIIII Steps to a Simulation

23. #1 State the problem or question #2 State the assumptions #3 Assign digits to represent outcomes #4 Simulate many repetitions #5 State conclusions Using our 20 repetitions, you have an estimate of the true chance of Ichiro having a multi-hit game. In my simulation, his chance was 9/20 or 45%. Steps to a Simulation

24. Simulations • This concludes this presentation.