I still don’t understand how to assign random digits for a simulation. For example, if the problem gives you something like “Bob makes freethrow shots 70% of the time, and his coach wants to calculate the likelihood of Bob making his free throws 4 out of the 5 times”. How would you assign digits to that type of problem??

Let each random digit represent one free throw. We could let a digit less than 7 represent making a free throw (note that this occurs with probability 7/10=70% since 7 of the 10 possible values from 0 to 9 are less than 7), and let a digit greater than or equal to 7 represent missing a free throw. Now generate a larger number of separate groups of 5 random digits. Count the number of groups, and also count how many of these groups contain 4 or more digits that are less than 7 (or equivalently no more than 1 digit that is greater than or equal to 7) . The number of groups containing 4 or more digits that are less than 7, divided by the number of groups, is an estimate of the likelihood (or probability) of making at least 4 out of 5 free throws. Example: let's look at the 16 groups of 5 random digits given in this lesson. 96565 05007 16605 81194 14873 04197 85576 45195 11169 15529 33241 83594 01727 86595 65723 82322 Out of these 16 groups, we find that 9 have 4 or more digits that are less than 7. So 9/16 is an estimate of the likelihood of making at least 4 out of 5 free throws. By the way, the theoretical likelihood is (5 choose 4)(0.7)^4 (0.3) + (5 choose 5)(0.7)^5 = 5*0.07203 + 1*0.16807 = 0.52822. So the estimate 9/16 = 0.5625 is not bad, considering that we used only 16 groups of 5 random digits.

So when assigning random digits, how do you know when to use double digits and when to use singe digits?

How many digits you use all depends on your sample size. If you are assigning digits to a sample of 100 people, then you'll need double digits, all numbers from 0 to 99. A general rule of thumb is to subtract your sample size by one and assign that many digits. So for a sample size of 20, you'll need the amount of digits in 19 (20 - 1), which is 2 digits. Hope this helped.

I don't understand the first and third one about choosing numbers between 1-90 and why more chances for the prizes?

it basically tries to say in a random sense what are the chances of 5 students to be assigned for cleaning duty. And in a random sense what are the chances of someone winning 6 prizes.

Is it just me or there is a typo in Problem 1 when the parameters are given? It says 'If the 2-digit number is anything between 01 and 90, that student is assigned LUNCH duty. Skip any other 2-digit number.' Then the question goes on an asks about CLEANING duty. I spent a good ten minutes thinking about how I'm supposed to solve it with no cleaning duty related information.. So it's a bit confusing.

Yeah mate. Cleaning duty or Lunch duty, I think they are talking about ignoring numbers 0 and 91-99.

How likely are three "1"s in sequence to appear in one table of random digits?

Assume every digit in the sequence is independent of each other, the digit generator is fair, so that would leave us with 3 digits in sequence. Your question is a bit vague, but let us assume we have a sequence of 10 digits. We know, that the probability of 3 digits in sequence is 1/10 * 1/10 * 1/10 = 1/1000. Now, let's look at the number of places in a sequence of 10 digits, where 3 ones can be placed. 111xxxxxxx x111xxxxxx ... xxxxxxx111 which leaves us with 8 possibilities. So in total, we have 8 times 1/1000 = 8/1000 or a probability of 1/125 so a bit less than 1%.

Main content

High school statistics

Course: High school statistics > Unit 6

Lesson 4: Probability from simulations

Simulation and randomness: Random digit tables

Google Classroom

We can simulate events involving randomness like picking names out of a hat using tables of random digits. Tables of random digits can be used to simulate a lot of different real-world situations. Here's

2

lines of random digits we'll use in this worksheet:

Line

1

96565 05007 16605 81194 14873 04197 85576 45195

Line

2

11169 15529 33241 83594 01727 86595 65723 82322

Things to know about random digit tables:

Each digit is equally likely to be any of the $10$ ‍ digits $0$ ‍ through $9$ ‍.
The digits are independent of each other. Knowing about one part of the table doesn't give away information about another part.
The digits are put in groups of $5$ ‍ just to make them easier to read. The groups and rows have no special meaning. They are just a long list of random digits.

Problem 1: Getting a random sample

There are

90

students in a lunch period, and

5

of them will be selected at random for cleaning duty every week. Each student receives a number

01 - 90

and the school uses a random digit table to pick the

5

students as follows:

Start at the left of Line $1$ ‍ in the random digits provided.
Look at $2$ ‍-digit groupings of numbers.
If the 2-digit number is anything between $01$ ‍ and $90$ ‍, that student is assigned lunch duty. Skip any other $2$ ‍-digit number.
Skip a $2$ ‍-digit number if it has already been chosen.

Line

1

96565 05007 16605 81194 14873 04197 85576 45195

Which $5$ ‍ students should be assigned cleaning duty?

Problem 2: Doing a simulation

A cereal company is giving away a prize in each box of cereal and they advertise, "Collect all

6

prizes!" Each box of cereal has

1

prize, and each prize is equally likely to appear in any given box. Caroline wonders how many boxes it takes, on average, to get all

6

prizes.

She decides to do a simulation using random digits as follows:

Start at the left of Line $2$ ‍ in the random digits provided.
Look at single digit numbers.
The digits $1 - 6$ ‍ represent the different prizes.
She ignores the digits $0, 7, 8, 9$ ‍.
One trial of the simulation is done when all $6$ ‍ digits have appeared.
At the end of the trial, she counts how many digits it took for every digit $1 - 6$ ‍ to appear (ignoring the other digits).

Line

2

11169 15529 33241 83594 01727 86595 65723 82322

question a

How many boxes of cereal did it take to get all $6$ ‍ prizes?

boxes

question b

Caroline did some more trials of her simulation. Each trial, she recorded how many boxes it took to get all

6

prizes. Her results are shown in the table below.

Trial #	Number of boxes
$1$ ‍	$12$ ‍
$2$ ‍	$17$ ‍
$3$ ‍	$15$ ‍
$4$ ‍	$7$ ‍
$5$ ‍	$20$ ‍

On average, how many boxes of cereal did it take Caroline to get all $6$ ‍ prizes?
If necessary, round your answer to the nearest tenth.

boxes

question c

Caroline's friend Grant did his own simulation. He did his just like Caroline, but he did

20

trials instead of

5

. On average, it took him

14.8

boxes to get all

6

prizes.

Whose results are more likely to give a closer estimate to the true average number of boxes it takes to get all $6$ ‍ prizes?

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Aarti Jain
Posted 7 years ago. Direct link to Aarti Jain's post “I still don’t understand ...”
I still don’t understand how to assign random digits for a simulation. For example, if the problem gives you something like “Bob makes freethrow shots 70% of the time, and his coach wants to calculate the likelihood of Bob making his free throws 4 out of the 5 times”. How would you assign digits to that type of problem??
Button navigates to signup pageComment on Aarti Jain's post “I still don’t understand ...”
(9 votes)
Answer
- Ian Pulizzotto
  Posted 7 years ago. Direct link to Ian Pulizzotto's post “Let each random digit rep...”
  Let each random digit represent one free throw. We could let a digit less than 7 represent making a free throw (note that this occurs with probability 7/10=70% since 7 of the 10 possible values from 0 to 9 are less than 7), and let a digit greater than or equal to 7 represent missing a free throw.
  
  Now generate a larger number of separate groups of 5 random digits. Count the number of groups, and also count how many of these groups contain 4 or more digits that are less than 7 (or equivalently no more than 1 digit that is greater than or equal to 7) . The number of groups containing 4 or more digits that are less than 7, divided by the number of groups, is an estimate of the likelihood (or probability) of making at least 4 out of 5 free throws.
  
  Example: let's look at the 16 groups of 5 random digits given in this lesson.
  
  96565 05007 16605 81194 14873 04197 85576 45195 11169 15529 33241 83594 01727 86595 65723 82322
  
  Out of these 16 groups, we find that 9 have 4 or more digits that are less than 7. So 9/16 is an estimate of the likelihood of making at least 4 out of 5 free throws.
  
  By the way, the theoretical likelihood is
  (5 choose 4)(0.7)^4 (0.3) + (5 choose 5)(0.7)^5 = 5*0.07203 + 1*0.16807 = 0.52822.
  
  So the estimate 9/16 = 0.5625 is not bad, considering that we used only 16 groups of 5 random digits.
  Comment on Ian Pulizzotto's post “Let each random digit rep...”
  (41 votes)
Aarti Jain
Posted 7 years ago. Direct link to Aarti Jain's post “So when assigning random ...”
So when assigning random digits, how do you know when to use double digits and when to use singe digits?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Owen S
  Posted 7 years ago. Direct link to Owen S's post “How many digits you use a...”
  How many digits you use all depends on your sample size. If you are assigning digits to a sample of 100 people, then you'll need double digits, all numbers from 0 to 99.
  A general rule of thumb is to subtract your sample size by one and assign that many digits. So for a sample size of 20, you'll need the amount of digits in 19 (20 - 1), which is 2 digits.
  
  Hope this helped.
  Comment on Owen S's post “How many digits you use a...”
  (9 votes)
Igor
Posted 6 years ago. Direct link to Igor's post “How could we solve Questi...”
How could we solve Question A of problem 2 without using random tables?
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(1 vote)
Answer
- Robb
  Posted 6 years ago. Direct link to Robb's post “Jerry's answer is great, ...”
  Jerry's answer is great, and following link is a wikipedia link to this concept
  https://en.wikipedia.org/wiki/Coupon_collector%27s_problem
  Button navigates to signup page
  (0 votes)
Darriel Meadows
Posted 3 years ago. Direct link to Darriel Meadows's post “I don't understand the fi...”
I don't understand the first and third one about choosing numbers between 1-90 and why more chances for the prizes?
Button navigates to signup pageComment on Darriel Meadows's post “I don't understand the fi...”
(2 votes)
Answer
- ARNAB DAS
  Posted 2 years ago. Direct link to ARNAB DAS's post “it basically tries to say...”
  it basically tries to say in a random sense what are the chances of 5 students to be assigned for cleaning duty. And in a random sense what are the chances of someone winning 6 prizes.
  Button navigates to signup page
  (2 votes)
9725952
Posted 3 months ago. Direct link to 9725952's post “I am still confused on ho...”
I am still confused on how to do the first 2 questions.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Saul Chavez
  Posted a month ago. Direct link to Saul Chavez's post “The reason I missed the f...”
  The reason I missed the first question the first time is because I included the number 96 when we were asked to just include the numbers 1 through 90. Does that help? For the second question you look at start at the beginning of the line of numbers and count from left to right until you have counted all the numbers 1 through 6. Afterward you count how many times it took you to get from 1 to 6 starting at the left (once again) and this time you're going to count all the numbers 1 through 6 (you're going to ignore any numbers above 6 because there were only 6 prizes that she needed to get) therefore once you count every number that it took (even counting repeating numbers) you come up with the answer of 12 boxes that it took. Does that make sense?
  Button navigates to signup page
  (1 vote)
zsofivamos
Posted 4 years ago. Direct link to zsofivamos's post “Is it just me or there is...”
Is it just me or there is a typo in Problem 1 when the parameters are given? It says
'If the 2-digit number is anything between 01 and 90, that student is assigned LUNCH duty. Skip any other 2-digit number.'
Then the question goes on an asks about CLEANING duty. I spent a good ten minutes thinking about how I'm supposed to solve it with no cleaning duty related information.. So it's a bit confusing.
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- ARNAB DAS
  Posted 2 years ago. Direct link to ARNAB DAS's post “Yeah mate. Cleaning duty ...”
  Yeah mate. Cleaning duty or Lunch duty, I think they are talking about ignoring numbers 0 and 91-99.
  Button navigates to signup page
  (3 votes)
denilsonfbar
Posted 3 years ago. Direct link to denilsonfbar's post “How likely are three "1"s...”
How likely are three "1"s in sequence to appear in one table of random digits?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- dashpointdash
  Posted 3 years ago. Direct link to dashpointdash's post “Assume every digit in the...”
  Assume every digit in the sequence is independent of each other, the digit generator is fair, so that would leave us with 3 digits in sequence.
  
  Your question is a bit vague, but let us assume
  we have a sequence of 10 digits.
  
  We know, that the probability of 3 digits in sequence is
  1/10 * 1/10 * 1/10 = 1/1000.
  
  Now, let's look at the number of places in a sequence of 10 digits, where 3 ones can be placed.
  
  111xxxxxxx
  x111xxxxxx
  ...
  xxxxxxx111
  
  which leaves us with 8 possibilities.
  
  So in total, we have 8 times 1/1000 = 8/1000 or
  a probability of 1/125 so a bit less than 1%.
  Comment on dashpointdash's post “Assume every digit in the...”
  (2 votes)
Sadiq Jama
Posted 7 years ago. Direct link to Sadiq Jama's post “What does random numbers ...”
What does random numbers means?
Button navigates to signup pageComment on Sadiq Jama's post “What does random numbers ...”
(2 votes)
Answer
Julia Robison
Posted 6 years ago. Direct link to Julia Robison's post “Why is the box number 14....”
Why is the box number 14.2 and not rounded to the nearest whole digit.
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(1 vote)
Answer
Zahara Sherif
Posted 7 years ago. Direct link to Zahara Sherif's post “I don't understand the in...”
I don't understand the instruction used for question A, is there another way to solve the Simulation problems?
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(1 vote)
Answer
- aleferman
  Posted 6 years ago. Direct link to aleferman's post “if you add all the number...”
  if you add all the number of boxes then divide with 6 is not the correct answer?
  Button navigates to signup page
  (1 vote)