From Digit Assignments to a Random Digit Table
In Assigning Random Digits to Outcomes, you matched equally likely digits or labels to outcomes in a chance model. A random digit table supplies the chance values for that model. To use it fairly, decide how to read the table before looking at the values, then follow the same rules throughout the simulation.
The digits in a table are usually printed in a long sequence. You can read across a row, down a column, or in another specified direction, but you must choose the direction in advance. If your model uses labels from 00 through 99, read the digits in groups of two. Keep leading zeros: the group 04 is the label 04, not 4.
A table may produce labels that do not represent any outcome in your model. These are unused values. You skip them and keep reading; you do not assign them to the closest outcome or change the digit grouping. Whether a repeated label is skipped depends on the chance process being simulated. In a process that selects distinct individuals without replacement, a repeat within that selection is ignored. In separate independent trials, a label may appear again in a later trial.
Set the Reading Rules Before Starting
Suppose a trial means selecting three different tickets from a set of 12 tickets, numbered 01 through 12. Tickets 01 through 04 are winners; tickets 05 through 12 are not. One trial’s event of interest is selecting at least one winning ticket.
Use two-digit groups because the ticket labels have two digits. The groups 01 through 12 represent tickets. The group 00 and groups 13 through 99 are unused, so skip them. Since the three tickets are selected without replacement, accept a ticket label only once in a trial. If a label repeats before the trial has three tickets, skip that repeat and read the next group.
- Choose the starting point and direction before reading any digits.
- Read consecutive groups of two digits without changing the grouping.
- Accept 01–12 as ticket labels; skip 00 and 13–99.
- Within a trial, skip a ticket label already accepted for that trial.
- End a trial after three distinct tickets are accepted, then begin the next trial with the next unread group.
The duplicate rule is limited to the current trial. Once a trial ends, its accepted labels are not reserved for the next trial. Each trial represents a fresh selection of three tickets from the full set. Reusing a label in a later trial is therefore allowed.
Worked Example: Decide Which Two-Digit Groups to Use
A simulation uses ticket labels 01 through 12. Read the following illustrative sequence in two-digit groups: 06, 00, 13, 06, 11, 08. In one trial, select two different tickets. Identify which groups are accepted or skipped, in order.
Apply the label range: The group 06 is in the range 01–12, so accept ticket 06. The group 00 is outside the range, so skip it. The group 13 is also outside the range, so skip it.
Apply the within-trial repeat rule: The next group, 06, is a repeat of the ticket already selected in this trial, so skip it. Accept 11 as the second ticket. The trial ends once two distinct tickets have been accepted; there is no need to use 08 for this trial.
The selected tickets are 06 and 11. This example shows why both rules matter: 00 and 13 are skipped because they are unused values, while the second 06 is skipped because it repeats a valid ticket within the same trial.
Run Five Trials in Sequence
To carry out several trials, keep reading forward through the digit table. Do not restart at the beginning for each trial or choose a new direction after seeing an inconvenient value. For each trial, mark the accepted labels and the reason any group is skipped. Stop that trial as soon as its required number of distinct labels has been accepted.
Here is a hypothetical excerpt from a random digit table, already divided into two-digit groups. It illustrates how to run five trials of the ticket simulation above. The chosen event is “at least one winning ticket,” where tickets 01–04 are winners. Each trial selects three distinct tickets.
Worked Example: Five Trials of Selecting Three Tickets
Read the groups in order, starting at the beginning of this illustrative excerpt and moving left to right. Accept labels 01–12, skip other values, and skip repeats within a trial. Stop each trial after three distinct tickets have been accepted.
| Trial | Groups read, in order | Three accepted tickets | At least one winner? |
|---|---|---|---|
| 1 | 07, 03, 03, 14, 01 | 07, 03, 01 | Yes |
| 2 | 12, 00, 05, 05, 08 | 12, 05, 08 | No |
| 3 | 04, 17, 02, 02, 10 | 04, 02, 10 | Yes |
| 4 | 09, 11, 13, 06 | 09, 11, 06 | No |
| 5 | 01, 04, 04, 00, 12 | 01, 04, 12 | Yes |
Trial 1: Accept 07 and 03. Skip the repeated 03. Skip 14 because it is outside the ticket range. Accept 01 as the third distinct ticket. Since 03 and 01 are winners, the event occurs.
Trial 2: Accept 12. Skip 00 because it is unused. Accept 05. Skip the repeated 05, then accept 08. None of the three selected tickets is numbered 01–04, so the event does not occur.
Trial 3: Accept 04, skip 17 because it is unused, and accept 02. Skip the repeated 02, then accept 10. Ticket 04 and ticket 02 are winners, so the event occurs.
Trial 4: Accept 09 and 11. Skip 13 because it is unused, then accept 06. None of these tickets is a winner, so the event does not occur.
Trial 5: Accept 01 and 04. Skip the repeated 04 and the unused value 00. Accept 12 as the third ticket. The event occurs because 01 and 04 are winners.
Summarize the five outcomes: The event occurred in trials 1, 3, and 5, for three occurrences in five trials. The event’s relative frequency in this simulation is \(3/5=0.60\).
The relative frequency of 0.60 describes these five simulated trials; it is not a claim that the event’s exact probability is 0.60. With only five trials, the result can vary noticeably from one run to another.
When Repeats Are—and Are Not—Skipped
A repeat is not automatically an error. The rule depends on what one trial represents. In the ticket example, each trial selects three different tickets without replacement. A ticket already selected cannot be selected again in that same trial, so its repeated label must be skipped. When the next trial begins, all 12 tickets are again available.
By contrast, suppose one trial represents a single independent check with two possible outcomes. A repeated label in a later trial may represent the same outcome again, and that is allowed. Independent trials do not require different outcomes or different random labels. Rejecting every label that appeared earlier in the simulation would change the process being modeled.
Worked Example: Repeated Labels Across Independent Trials
A device passes a check with probability 0.35. Use labels 00–34 for “pass” and 35–99 for “does not pass,” as in the digit-assignment method from the earlier tutorial. Suppose five successive two-digit groups from a random digit table are 08, 72, 08, 35, and 99. Simulate the five checks.
Apply the mapping in order: The label 08 represents pass. The label 72 represents does not pass. The next 08 again represents pass. The label 35 is in the 35–99 range, so it represents does not pass; the boundary label 35 is not in the pass range. The label 99 also represents does not pass.
| Trial | Label | Simulated outcome |
|---|---|---|
| 1 | 08 | Pass |
| 2 | 72 | Does not pass |
| 3 | 08 | Pass |
| 4 | 35 | Does not pass |
| 5 | 99 | Does not pass |
The repeated 08 is used again because these are separate checks, not a selection of distinct devices from one group. Two of the five simulated checks pass, so the relative frequency of passing is \(2/5=0.40\). Skipping the second 08 would incorrectly alter the sequence of independent outcomes.
Common Mistakes and AP Exam Tips
- Changing group size partway through. If the model uses labels 00–99, keep reading groups of two. Do not regroup later values to make a label fit.
- Dropping leading zeros. Read 04 as 04. It is a valid label from 01 through 12; it is not a one-digit value outside the stated format.
- Treating unused values as outcomes. For labels 01–12, values such as 00 and 13 are skipped. State the valid range and keep reading until the trial is complete.
- Skipping every repeated value in the entire simulation. Skip a repeat only when the trial models sampling without replacement and that label has already been selected in the current trial. A new trial starts a new selection.
- Restarting or choosing a new direction after an inconvenient sequence. Fix the start and direction before reading the table. Continue forward so that the chance process is not influenced by the results you see.
- Calling a small-run relative frequency the exact probability. A simulation’s relative frequency is an estimate based on its trials. With five trials, explain the observed count and frequency without claiming they equal the model probability.
For a full-credit description, identify the group size and reading direction, define which labels are accepted, state what happens to unused values and repeats, and explain when each trial ends. When reporting results, give the number of times the event occurred and divide by the total number of trials. Those details make the simulation reproducible and show that its rules match the process being modeled.
Check Your Understanding
Answer each question using the stated simulation rules.
- A model uses labels 01–18 and reads two-digit groups. Which of these are unused: 00, 09, 18, 19, and 88?
- In one trial, three different members are selected without replacement. If the same valid label appears twice before three members are selected, what should you do with the repeat?
- A new trial begins after the three-member selection is complete. May a label from the earlier trial be selected again in the new trial? Explain.
- Five trials produce the event outcomes yes, no, yes, no, yes. What is the event’s relative frequency in these trials?
- Why should you choose the table’s starting point and reading direction before examining its digits?