Unit 4 Randomness in Data

Unit 4 Randomness in Data
Topic 14 Probability (page 303)

OVERVIEW You have been studying methods for analyzing data, from displaying them graphically to describing them verbally and numerically. Earlier, you saw how to create your own samples of data by sampling from a larger population. Now we turn our attention to drawing inferences about the population based on a sample. As you learned earlier, this inference process is feasible only if you have randomly selected the sample from the population.

OVERVIEW At first glance it might seem that introducing randomness into the process would make it more difficult to draw reliable conclusions. Instead, you will find that randomness produces patterns that allow us to quantify how close the sample will come to the population result. This topic introduces you to the idea of probability and asks you to explore some of its properties.

Question number 7 assumes that you are playing solitaire by the rules.
Do the Preliminaries (page 304) Question number 7 assumes that you are playing solitaire by the rules.

What are the properties of probability?
Essential Question What are the properties of probability?

Activity 14-1 Random Babies (pages 305 to 308)
When it is not feasible to actually carry out an activity to investigate what would happen in the long run, simulation is used instead. Simulation is an artificial representation of a random process used to study its long-term properties.

Modified Directions! Take 2 sheets of scrap paper.
Cut one (1) into 4 equal sections and write a baby’s first name on each section. Divide, but do NOT cut the second sheet into 4 equal sections and write a mother’s last name in each section. Shuffle the babies names and randomly place them on the mother’s names. Record the requested information.

yes or no (a) Did Mrs. Johnson get the right baby? ____________ Did all mothers get the wrong baby? ____________ How many mothers got the right baby? __________ (b) yes or no 0 to 4 Part (a) is repetition #1. Repeat this 4 more times. Shuffle the babies names and randomly place them on the mother’s names for each repetition. Record the requested information.

1 2 3 4 5 Repetition # # of matches Johnson match? (use Y or N)
(b) Repetition # 1 2 3 4 5 Johnson match? (use Y or N) All wrong? # of matches (use 0 to 4)

N Y 1 2 3 4 5 Repetition # # of matches Johnson match? (use Y or N)
(b) Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4)

Johnson matches? All wrong? (c) 1 5 2 10 3 15 4 20 25 6 30 7 35 8 40 9
student cum reps of these 5 cum tot cum prop 1 5 2 10 3 15 4 20 25 6 30 7 35 8 40 9 45 50 11 55 12 60 13 65 14 70 75 16 80 17 85 18 90 19 95 100

Count the number of your Johnson matches!
Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) Enter 3 for “Of these 5”.

Johnson matches? All wrong? (c) 1 5 3 .60 2 10 15 4 20 25 6 30 7 35 8
student cum reps of these 5 cum tot cum prop 1 5 3 .60 2 10 15 4 20 25 6 30 7 35 8 40 9 45 50 11 55 12 60 13 65 14 70 75 16 80 17 85 18 90 19 95 100

Count the number of your “All wrong?”!
Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) Enter 2 for “Of these 5”.

Johnson matches? All wrong? (c) 1 5 3 .60 2 .40 10 15 4 20 25 6 30 7
student cum reps of these 5 cum tot cum prop 1 5 3 .60 2 .40 10 15 4 20 25 6 30 7 35 8 40 9 45 50 11 55 12 60 13 65 14 70 75 16 80 17 85 18 90 19 95 100

.28 .33 (d) This is just an example of what your graph may look like.
The proportion of trials where Johnson matches appears to be approaching ____. The proportion with no matches appears to be approaching _____. .33

The probability of a random event is the
long-run proportion (or relative frequency) of times the event would occur if the random process were repeated over and over under identical conditions. One can approximate a probability by simulating the process a large number of times. Simulation leads to an empirical estimate of the probability. Empirical results are based on experience, observations, or experimentation rather than theory or pure logic.

1 2 3 4 1.00 total # of matches count proportion
(f) Combine your results with the rest of the class. Please note that you will be entering 5 tally marks. # of matches 1 2 3 4 total count proportion 1.00

I will enter 2 tally marks for the two 0 matches.
Tally the # of 0 matches from part (b). Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) I will enter 2 tally marks for the two 0 matches.

Enter the 2 tally marks for the # of 0 matches.
1 2 3 4 total count || proportion 1.00

Tally the # of 1 matches from part (b). Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) I enter NO tally marks since there are none.

Enter NO tally marks for the # of 1 matches.

I will enter 2 tally marks for the two 2 matches.
Tally the # of 2 matches from part (b). Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) I will enter 2 tally marks for the two 2 matches.

Enter the 2 tally marks for the # of 2 matches.

Tally the # of 3 matches from part (b). Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) I enter NO tally marks since there are none.

Enter NO tally marks for the # of 3 matches.

I will enter 1 tally mark for the one 4 matches.
Tally the # of 4 matches from part (b). Repetition # 1 2 3 4 5 Johnson match? (use Y or N) N Y All wrong? # of matches (use 0 to 4) I will enter 1 tally mark for the one 4 matches.

1 2 3 4 1.00 total PLEASE TAKE NOTE THAT 5 TALLY MARKS ARE ENTERED!
Enter the 1 tally mark for the # of 4 matches. # of matches 1 2 3 4 total count || | proportion 1.00 PLEASE TAKE NOTE THAT 5 TALLY MARKS ARE ENTERED!

1 2 3 4 1.00 total # of matches count proportion
(f) Now enter your data on the board. Remember, you will be entering 5 tally marks. # of matches 1 2 3 4 total count proportion 1.00

(g) In __________ of these simulated cases, at least one mother got the correct baby.
My empirical estimate of the probability of no matches is __________. My empirical estimate of the probability of at least one match is __________.

(j) An outcome of exactly 3 matches is impossible because …
(k) Is it impossible to get four matches? _________ Would you call it rare? _________ Would you call it unlikely? _________ Would you call a result of 0 matches, or 1 match, or 2 matches, to be unlikely? _________ if there are 3 matches, then all 4 must match. NO YES Yes or No NO

What are empirical estimates of probabilities?
Essential Question What are empirical estimates of probabilities?

Activity 14-2 Random Babies (continued) (pages 309 to 311)
In situations where the outcomes of a random process are equally likely, exact probabilities can be calculated by listing all of the possible outcomes and counting the proportion that correspond to the event of interest. The listing of all possible outcomes is called the sample space .

Random Babies Sample Space
(a) There are _________ different arrangements for returning the 4 babies to their mothers. 24

How to get that without all the writing.
There are 4 different babies going to 4 different mothers. Order really matters! If you don’t believe me, ask any new mother. This is a permutation, which is an arrangement of “r” objects from a set of “n” objects. nPr = P(n,r) ! means FACTORIAL.

This is an arrangement of “4” babies from a set of “4” babies.
Permutation This is an arrangement of “4” babies from a set of “4” babies. 4P4 = P(4,4) 0! = 1 You don’t believe me? Ask your calculator.

To enter this on your calculator,
Permutation To enter this on your calculator, follow this sequence … Enter 4 then press… MATH → PRB ↓ nPr enter 4 then press ENTER 4 nPr 4 24

For Your Information (FYI)
If order did NOT matter. This is would be a combination, which is also an arrangement of “r” objects from a set of “n” objects. nCr = C(n,r) 4C4 = 1 … remember order doesn’t matter.

(b) matches matches matches matches matches matches 1234:_______ :_______ :_______ :_______ :_______ :_______ 2134:_______ :_______ :_______ :_______ :_______ :_______ 3124:_______ :_______ :_______ :_______ :_______ :_______ 4123:_______ :_______ :_______ :_______ :_______ :_______ (c) In how many arrangements is the number of “matches” equal to exactly: 4:_______ 3:_______ 2:_______ 1:_______ 0:_______ 4 2 2 1 1 2 2 1 1 1 2 1 1 1 2 1 6 8 9 = 24

(c) In how many arrangements is the number of “matches” equal to exactly:
4:_______ 3:_______ 2:_______ 1:_______ 0:_______ Probabilities for the corresponding numbers: [comment] 1 6 8 9 Divide each number by 24. .04 .25 .33 .38 The exact probabilities should be fairly close to the simulated results.

An empirical estimate from a simulation
generally gets closer to the actual probability as the number of repetitions increases . The graph that represents __________ times produces empirical estimates closest to the actual probabilities. Show your work for the calculation of the average (mean) number of matches per repetition of the process. mean = ________ 10,000 4( ) + 3(0) + 2( ) + 1( ) + 0( )

The long - run average value achieved by a numerical random
process is called its expected value . To calculate this expected value from the (exact) probability distribution, multiply each outcome by its probability, and then add these up over all of the possible outcomes.

This should be reasonably close to the simulated number.
(g) [show your work] expected number of matches =________ [compare] 4(.04)+3(0)+2(.25)+1(.33)+0(.38) = .99 1 This should be reasonably close to the simulated number.

Activity 14-3 Weighted Coins (pages 311 to 313)
The probabilities of landing heads for the six coins are: coin A coin B ⅓ coin C coin D coin F coin E ⅘

(a) Take your guesses! rep 1 H T 2 3 4 5 1st coin 2nd coin 3rd coin
4th coin 5th coin 6th coin 1 H T 2 3 4 5 relative frequency coin guess (letter)

(a) Take your guesses! 0.8 1.0 0.2 rep 1 H T 2 3 4 5 1st coin 2nd coin
3rd coin 4th coin 5th coin 6th coin 1 H T 2 3 4 5 relative frequency 0.8 1.0 0.2 coin guess (letter)

NO Are you confident that all of your guesses are correct? ________ [explain] The confidence of your guesses is most likely hindered by the duplicated frequencies.

(c) Again, take your guesses!
1st coin 2nd coin 3rd coin 4th coin 5th coin 6th coin relative frequency 0.70 0.90 0.20 0.80 1.00 coin guess (letter)

(d) And again, take your guesses!
1st coin 2nd coin 3rd coin 4th coin 5th coin 6th coin relative frequency 0.56 0.88 0.28 1.00 0.20 coin guess (letter) n = 50 1st coin 2nd coin 3rd coin 4th coin 5th coin 6th coin relative frequency 0.58 0.92 0.26 0.78 1.00 0.32 coin guess (letter)

After a total of 50 flips, are you reasonably
confident that your guesses are correct? ________ [explain] yes / no The confidence that your guesses are correct should increase as the number of flips increase. The frequencies should be getting closer to the actual probabilities.

Here are the correct answers.
1st coin 2nd coin 3rd coin 4th coin 5th coin 6th coin relative frequency 0.58 0.92 0.26 0.78 1.00 0.32 coin guess (letter) C E A D F B coin C coin A coin F coin E ⅘ coin D coin B ⅓

long short stabilize (level off)
(f) Probability describes the ________ - term behavior of random processes and not the _______ - term behavior because many trials are needed before an accurate probability can be deduced. For instance, this graph shows a lot of variation in relative frequencies near _____ repetitions. However, as the six coins are flipped more often, the relative frequencies _______________________ and show us the long-term behavior we are interested in. short stabilize (level off)

Word Master Activity Worksheet
Assignment Activity 14-6: Random Cell Phones (page 318 & 319) Assignment Activity 14-7: Equally Likely Events (page 319) Word Master Activity Worksheet What are the properties of probability?

Activity 14-4 Boy and Girl Births (pages 314 to 316)
For simplicity we will assume that the probability of a boy birth is 50% independently from child to child.

Use Table I: Random Number Table on page 587
Use Table I: Random Number Table on page Pick a line, write down the first 4 digits, then let even # = girl and odd # = boy. Family 1 Child number 1 2 3 4 # girls Random Digit Gender

I am using line # 7. Family 1 Child number 1 2 3 4 # girls
Random Digit 8 5 9 Gender

even # = girl and odd # = boy
I am using line # 7. Family 1 Child number 1 2 3 4 # girls Random Digit 8 5 9 Gender G B even # = girl and odd # = boy

even # = girl and odd # = boy
I am using line # 7. Family 1 Child number 1 2 3 4 # girls Random Digit 8 5 9 Gender G B 1 even # = girl and odd # = boy

Repeat this for a total of 5 families, recording the number of girls in each family. even # = girl and odd # = boy Family 1 Child number 1 2 3 4 # girls Random Digit Gender Family # 1 2 3 4 5 # of girls

I am continuing on line # 7.
Family 1 Child number 1 2 3 4 # girls Random Digit 8 5 9 Gender G B 1 Family # 1 2 3 4 5 # of girls

number of simulated families Empirical probability
Combine your results with the rest of the class, then divide by the total of simulated families to get the empirical results. Everyone will enter 5 tally marks! Family # 1 2 3 4 5 # of girls # of girls in family 1 2 3 4 number of simulated families Empirical probability

Combine your results with the rest of the class, then divide by the total of simulated families to get the empirical results. Everyone will enter 5 tally marks! Family # 1 2 3 4 5 # of girls # of girls in family 1 2 3 4 number of simulated families | Empirical probability

Combine your results with the rest of the class, then divide by the total of simulated families to get the empirical results. Everyone will enter 5 tally marks! Family # 1 2 3 4 5 # of girls # of girls in family 1 2 3 4 number of simulated families | || Empirical probability

Enter your 5 tally marks! # of girls in family 1 2 3 4 number of simulated families | || Empirical probability

In situations like this where each trial has only two possible outcomes and the probabilities remain the same on each trial independently from trial to trial, exact probabilities can be calculated from the binomial distribution .

Sample Space for a Family with 4 Children
0 girls 1 girls 2 girls 3 girls 4 girls BBBB 4C0 = 1

0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB BGBB BBGB BBBG 4C1 = 4

0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB BGBB GBGB BBGB GBBG BBBG BGGB BGBG BBGG 4C2 = 6

0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB GGGB BGBB GBGB GGBG BBGB GBBG GBGG BBBG BGGB BGGG BGBG BBGG 4C3 = 4

0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB GGGB GGGG BGBB GBGB GGBG BBGB GBBG GBGG BBBG BGGB BGGG BGBG BBGG n = 16 4C4 = 1

# of girls in family 1 2 3 4 Exact probabilities .0625 0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB GGGB GGGG BGBB GBGB GGBG BBGB GBBG GBGG BBBG BGGB BGGG BGBG BBGG

# of girls in family 1 2 3 4 Exact probabilities .0625 0.25 0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB GGGB GGGG BGBB GBGB GGBG BBGB GBBG GBGG BBBG BGGB BGGG BGBG BBGG

# of girls in family 1 2 3 4 Exact probabilities .0625 0.25 0.375 0 girls 1 girls 2 girls 3 girls 4 girls BBBB GBBB GGBB GGGB GGGG BGBB GBGB GGBG BBGB GBBG GBGG BBBG BGGB BGGG BGBG BBGG

# of girls in family 1 2 3 4 Exact probabilities .0625 0.25 0.375 # of girls in family 1 2 3 4 number of simulated families Empirical probability (d) [comment] The class results should be fairly close to the exact probabilities.

According to the exact probabilities, is it more likely for a family to have two children of the same gender or three children of the same gender? or 3 [explain] The probability of having 2 children of the same gender is 37.5%. There are 2 ways to accomplish having 3 children of the same gender, … 1 girl and 3 boys or 3 girls and 1 boy. Therefore, the probability of having 3 children of the same gender is 50%.

Do you expect the likelihood of an exact 50/50 gender split to increase or to decrease with larger families? increase or decrease [explain] Answer this question as you believe it should be answered. Please, completely explain your answer. I will not take points off if you are wrong.

Follow the directions to do the calculations on your graphing calculator.
CHOOSE A PARTNER TO RUN THE SIMULATION WITH 4 CHILDREN AND THE OTHER RUN THE SIMULATION WITH 10 CHILDREN. YOU WILL NEED A SIGNIFICANT AMOUNT OF MEMORY ON YOUR CALCULATOR TO DO RUN THIS SIMULATION!!! PLEASE REMOVE ANY UNNECESSARY LISTS BEFORE RUNNING THE THE SIMULATION.

Press… MATH → PRB ↓ 7:randBin( Enter … 4 (or 10) , .5 , 500 )
then press ENTER randBin(4,.5,500) # of children 4 or 10 # of observations to simulate the # of families probability

randBin(4,.5,500) {3 1 3 2 2 2 1 … Store into a list by using the
{ … Store into a list by using the STO key.

randBin(4,.5,500) { … AnsL1 Make a histogram and adjusting the WINDOW to Xmin=0, Xmax=5 (or 11), and Xscl=1, then press GRAPH.

Look at the bar for 2 children. Look at the bar for 5 children.
The proportion of the 4-children families that have 2 boys and 2 girls is … ___________. (exact = 37.5%) The proportion of the 10-children families that have 5 boys and 5 girls is … (exact = 24.6%) My expectation in (f) was __________________. Look at the bar for 2 children. Look at the bar for 5 children. confirmed or refuted

Assignment Activity 14-8: Interpreting Probabilities (page 319) What are the properties of probability?

Activity 14-5 Hospital Births (pages 316 & 317)
(a) Proportion of days of days that hospital A observed an equal count of girls is ________. Proportion of days of days that hospital B observed an equal count of girls is ________. The ______________ hospital has more days with an exact 50/50 split. 25% 90 ÷ 365 = 24.7% 13% 46 ÷ 365 = 12.6% smaller

A B Hosp. A is 143 days and Hosp. B is 31 days
(b) Hospital ______ has more days on which 60% or more of the births are girls. (births ≥ 60%) Is your prediction from the “Preliminaries” section supported? ________ Hospital ______ has more days on which between 41% and 59% of the births are girls. (40% < births < 60%) Hosp. A is 143 days and Hosp. B is 31 days yes / no B Hosp. A is 163 days and Hosp. B is 314 days

larger (d) While a __________ sample size makes it less likely to get an exact 50/50 split in the observed counts, the probability of getting a sample proportion close to one-half increases with a __________ sample. larger

symmetric and mound shaped.
This activity reveals that while a larger sample size makes it less likely to get an exact 50/50 split in the observed counts, the probability of getting a sample proportion close to ½ increases with a larger sample. Consequently, we are less likely to obtain a sample proportion far away from the long-term probability of ½. Also note that a larger sample produces a probability distribution that is quite symmetric and mound shaped.

WRAP-UP This topic has initiated your study of randomness by introducing you to the concept of probability. You have learned that probability is a long-run property of events, and you have studied probability by conducting simulations. You have carried out these simulations using physical devices such as index cards, using a table of random digits, and using your calculator. You have also studied probability more theoretically, through the notion of equally likely events, sample space, and expected value.

Your topic is due! What are the properties of probability?
Assignment Activity 14-9: Racquet Spinning (pages 319 & 320) We will do part (d) to this activity as a class. Everyone will get a chance to SPIN a racquet! Tennis players will take the lead. Your topic is due! What are the properties of probability?

Unit 4 Randomness in Data

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