1.Students Will Gather Into Equal Groups

Written by Bryan, Carl, and

Activity for 5.1:

1.Students will gather into equal groups

2. Students will answer questions

3. One student from each group will write the answer on the whiteboard

4. The first student to write down the correct answer on the whiteboard gains 2pts

5. Once all questions are answered, the group with the most points will win the game

Notes for 5.1:

Law of Large Numbers:If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value (the probability of that outcome).

Probability: Any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

*Outcomes that never occur have probability 0. Outcomes that occur every repetition has probability 1

Simulation: The imitation of chance behavior, based on a model that accurately reflects the situation.

*Performing a Simulation

STATE= What is the question of interest about some chance process?

PLAN= Describe how to use a chance device to imitate one repetition of the process.

Explain clearly how to identify the outcomes of the chance process and what variable to measure.

Do= Perform many repetitions of the simulation.

Conclude= Use the results of you simulation to answer the question of interest.

Questions:

1. Carl has a probability of falling down the stairs at 0.01. Interpret this probability as a long-run relative frequency.

◦(Answer)= Carl has a 1% chance of falling down the stairs

2. Bryan has a probability 0.60 of hitting a home run. Explain how you would use a six sided die to simulate one home run.

(Answer)= Let 1,2, and 3 represent Bryan making a home run, and 4 representing a miss. If 5 or 6 come up, ignore them and role again.

3. A roulette wheel has 38 colored slots (18 red, 18 black, 2 green). Make a simulation design.

(Answer)= Let 00-18 be red, let 19-37 be black, let 38 and 39 be green)

4. Determine whether each of the following simulation designs is valid. (YES or NO (Explain why))

According to a MHS poll, 75% of students eat lunch. To simulate choosing a random sample of 100 MHS students, roll a 4 sided die 100 times. A result of 1,2 or 3 means the student eats lunch, a 4 means the student doesn't eat lunch.

YES. The chance of rolling a 1,2 or 3 is 75% on a 4 sided die, and the rolls of independent of each other.

Mr. Robertson will randomly select students in the hallway and make sure they have a pass. Last week Mr. Robertson randomly selected 76 students, 12 of those students were underclassmen, and 64 were upperclassmen. Some students were confused when none of the 10 randomly selected students were underclassmen.

(a)State the question of interest using the language of probability.
(b)How would you use random digits to imitate one repetition of the process?What variable would you measure?
(c)Use the line of random digits below to perform one repetition

7148709984290771486361683470526222451025

(d)In 100 repetitions of the simulation,there were 15 times when none of the 10 passengers chosen was seated in first class.What conclusion would you draw?

◦(Answer)=(a)What is the probability that in a random selection of 10 students, no underclassmen are chosen?

◦(Answer)=(b)Number the underclassmen 01–12 and the upperclassmen 13–76. Ignore all other numbers. Look up two-digit numbers (SHEET) until you have 10 unique numbers. Count the number of two-digit numbers between 01 and 12.

◦(Answer)=(c)Sample is 71 48 70 29 07 63 61 68 34 52. There is one student among the 10 selected who is an underclassmen in this sample.

◦(Answer)=(d)Since in 15% of the upperclassmen was chosen, it seems plausible that the actual selection was random.

What's the probability that in a randomly selected group of 30 unrelated teachers, at least two have the same birthday? Let's make two assumptions to simplify the problem. First, we'll ignore the possibility of a Feburary 29 birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining 365 days of the year.

(a)How would you use random digits to imitate one repetition of the process?What variable would you measure?
Use technology to perform 5 repetitions. Record the outcome of each repetition.
Would you be surprised to learn that the theoretical probability is 0.71%? Why or why not?

(a)Read off 30 three-digit numbers from the table,ignoring numbers greater than 365 and 000.Record whether there were any repeats in the sample or not.

(b)Answers will vary.We used Minitab to select five samples.There were repeats in all five samples.
(c)Answers will vary.After the simulation we would not be surprised that the probability is 0.71 since we found repeats in 100% of our samples.

About 7% of seniors in Madison High School have dome form of “SENIORITIS” Suppose we randomly select one senior from MHS at a time until we find one who has senioritis. How many seniors would we expect to choose, on average? Design and carry out a simulation to answer this problem. (FOLLOW THE 4-STEP PROCESS)

State:How many seniors would we expect to choose in order to find one who has senioritis?

Plan: Use technology to simulate choosing seniors. Label number 01-07 as seniors with senioritis and all other two-digit number as seniors without senioritis. Use technology to produce two-digit numbers until a number between 01 and 07 appears. Count how many two-digit numbers there are in the sample.

Do: We did 50 repetitions of the simulation using technology. The first repetition is 17 33 49 41 02. The number in bold is the stopping point. For this repetition we chose 5 seniors in order to get the one senior with senioritis. In 50 repetitions, the average number of men needed was 16.88.
Conclude:Based on our simulation,we would suggest that we would need to sample about 17 seniors, on average.

You read in a textbook about a poker, that said the probability that each of the four players is dealt exactly one ace is about 0.11. This means that ?

(a)in every 100 poker deals each player has one ace exactly 11 times.
(b) in one million poker deals, the number of deals on which each player has one ace will be exactly 110,000.
(c) in a very large number of poker deals, the percent of deals on which each player has one ace will be very close to 11%.
(d) in a very large number of poker deals, the average number of aces in a hand will be very close to 0.11.
(ANSWER)= C

An APSTAT Poll asked whether students experienced stress “a lot of the day yesterday.” 40% said they did. The APSTAT Poll report said “Results are based on telephone interviews with 250 students.

(a) Identify the population of the sample
(b) Explain how under-coverage could lead to bias in this survey.

(ANSWER)= (A)= The population is MHS students, and the sample is 250 students who interviewed.

(ANSWER)=(B)= Since the interviews were conducted using the telephone, those people who do not have a telephone were excluded from the APSTAT Poll. In general, people who do not have phones tend to have a lower IQ, and may experience more stress in their lives than the population of students as a whole.

You want to estimate the probability that the player makes 5 or more of 10 free throws. You simulate 10 shots, 25 times, and get the following numbers of hits:

5-7-5-4-1-5-3-4-3-4-5-3-4-4-6-3-4-1-7-4-5-5-6-5-7

What is your estimate of the probability?

(a) 5/25 or 0.20
(b)11/25 or 0.44
(c) 12/25 pr 0.48
(d) 16/25 or 0.64
(e) 19/25 or 0.76

(ANSWER)=C

Carl has a probability of falling down the stairs at 0.01. Interpret this probability as a long-run relative frequency.

Bryan has a probability 0.60 of hitting a home run. Explain how you would use a six sided die to simulate one home run.

A roulette wheel has 38 colored slots (18 red, 18 black, 2 green). Make a simulation design.

Determine whether each of the following simulation designs is valid. (YES or NO)

(Explain why)

5. Mr. Robertson will randomly select students in the hallway and make sure they have a pass. Last week Mr. Robertson randomly selected 76 students, 12 of those students were underclassmen, and 64 were upperclassmen. Some students were confused when none of the 10 randomly selected students were underclassmen.

(a)State the question of interest using the language of probability.
(b)How would you use random digits to imitate one repetition of the process?What variable would you measure?
(c)Use the line of random digits below to perform one repetition

7148709984290771486361683470526222451025

(d)In 100 repetitions of the simulation,there were 15 times when none of the 10 passengers chosen was seated in first class. What conclusion would you draw?

6. What's the probability that in a randomly selected group of 30 unrelated teachers, at least two have the same birthday? Let's make two assumptions to simplify the problem. First, we'll ignore the possibility of a February 29 birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining 365 days of the year.

(a)How would you use random digits to imitate one repetition of the process?What variable would you measure?
(b)Use technology to perform 5 repetitions. Record the outcome of each repetition.
(c)Would you be surprised to learn that the theoretical probability is 0.71%? Why or why not?

About 7% of seniors in Madison High School have dome form of “SENIORITIS” Suppose we randomly select one senior from MHS at a time until we find one who has senioritis. How many seniors would we expect to choose, on average? Design and carry out a simulation to answer this problem. (FOLLOW THE 4-STEP PROCESS)

8. You read in a textbook about a poker, that said the probability that each of the four players is dealt exactly one ace is about 0.11. This meanthat?

(a)in every 100 poker deals each player has one ace exactly 11 times.
(b) in one million poker deals, the number of deals on which each player has one ace will be exactly 110,000.
(c) in a very large number of poker deals, the percent of deals on which each player has one ace will be very close to 11%.
(d) in a very large number of poker deals, the average number of aces in a hand will be very close to 0.11.

9. An APSTAT Poll asked whether students experienced stress “a lot of the day yesterday.” 40% said they did. The APSTAT Poll report said “Results are based on telephone interviews with 250 students.

(a) Identify the population of the sample
(b) Explain how under-coverage could lead to bias in this survey.

10. You want to estimate the probability that the player makes 5 or more of 10 free throws. You simulate 10 shots, 25 times, and get the following numbers of hits:

5-7-5-4-1-5-3-4-3-4-5-3-4-4-6-3-4-1-7-4-5-5-6-5-7

What is your estimate of the probability?

(a) 5/25 or 0.20
(b)11/25 or 0.44
(c) 12/25 pr 0.48
(d) 16/25 or 0.64
(e) 19/25 or 0.76