A Fixed-Length Trial Has a Clear Boundary
In Simulating Repeated Trials Until Success, a trial continued until a specified outcome occurred. A guessing quiz works differently: the number of questions is fixed in advance. One trial represents one complete quiz, and it always uses exactly ten generated digits, whether the student guesses many answers correctly, only a few, or none.
This difference determines how to read the random outcomes. For a fixed-length trial, do not stop when a correct answer appears, and do not keep generating answers after question 10. Instead, read ten digits, count the correct answers among them, record that count as the trial’s score, and then begin the next trial with the next digit.
The count is the outcome of interest—not the identity or order of the questions answered correctly. For example, if the correct-answer digit appears three times among a trial’s ten digits, the simulated score is 3 correct, regardless of which question positions those appearances represent.
Build the Quiz Model with Random Digits
Suppose each quiz question has ten answer choices, labeled 0 through 9, and exactly one choice is correct. A student guesses randomly on every question. To simulate the student’s guess, generate one random digit for that question. For convenience, suppose choice 6 is the correct answer on every question. Then a generated 6 represents a correct guess, and any other digit represents an incorrect guess.
This digit assignment models a random guess because all ten digits are equally likely, and exactly one of them represents a correct answer. The model also assumes that the guesses on different questions are independent: the digit for one question does not affect which digit is generated for another. These assumptions describe the chance process being imitated; they do not claim that every real student guesses randomly or that every actual quiz has ten choices.
This is an application of the digit-assignment idea from Assigning Random Digits to Outcomes. One digit represents one question because there are ten equally likely choices and ten digits, 0 through 9. The assignment stays the same for all ten questions and across all simulated quizzes.
To generate one trial on a calculator, use randInt(0,9,10). This produces ten digits from 0 through 9. Count the number of 6s in the output. To simulate additional trials, generate another group of ten digits for each new quiz, or use a longer stream and divide it into consecutive groups of ten.
Use 6 for a correct answer and 0–5 and 7–9 for incorrect answers.
Use exactly ten digits for one 10-question quiz.
Read one random digit for each question, in order.
Count the 6s in the ten digits. That count is the number correct, from 0 to 10.
Use the next ten digits and make a new count. Do not carry a score or question over from the earlier trial.
Work Through One Complete Quiz
Suppose the ten generated digits are 2, 6, 1, 8, 6, 4, 0, 6, 3, 9. Each position represents the next question: the first digit is the guess for question 1, the second is the guess for question 2, and so on. There are three 6s, so the simulated quiz score is 3 correct out of 10.
Worked Example: Count Correct Guesses in One Trial
A student guesses on a ten-question quiz with ten answer choices per question. Digit 6 represents a correct guess. One trial produces the digits 4, 6, 2, 0, 6, 8, 1, 9, 3, 5. Record the trial outcome.
There are ten digits, so the trial includes all ten questions. The digit 6 appears in positions 2 and 5, and no other position contains a 6. Therefore, the student answered 2 questions correctly. The trial outcome is a score of 2 correct answers out of 10.
Notice that the trial does not end after the first 6 or after the student has reached any particular score. A 6 simply marks one correct answer. The fixed boundary is the tenth question.
Read a Stream as Consecutive Groups of Ten
A longer stream can represent several quizzes. Start at the first digit, divide the stream into consecutive groups of ten, and count the correct-answer digit in each group separately. The first ten digits make trial 1, the next ten make trial 2, and the next ten make trial 3. This fixed grouping is what prevents digits from one simulated quiz from being mistakenly included in another.
Worked Example: Record Scores for Three Trials
Use 6 to represent a correct answer. The generated stream is shown below. Divide it into three groups of ten digits and record the number correct in each simulated quiz.
Trial 1 contains three 6s, in positions 1, 3, and 6, so its score is 3. Trial 2 contains three 6s, in positions 2, 3, and 8, so its score is also 3. Trial 3 contains seven 6s, in positions 1 through 5, 7, and 9, so its score is 7.
| Trial | Number of digits | Number of 6s | Recorded score |
|---|---|---|---|
| 1 | 10 | 3 | 3 correct |
| 2 | 10 | 3 | 3 correct |
| 3 | 10 | 7 | 7 correct |
Each row represents one completed quiz. The two trials with scores of 3 are still separate trials: they happened to have the same recorded count, but they used different groups of digits. Trial 3 begins only after all ten digits in trial 2 have been used.
When using a random digit table instead of a calculator, keep the same boundaries. As described in Using a Random Digit Table, decide how to read the table before beginning. For this model, read one digit per question and use consecutive groups of ten. Do not change direction or regroup the digits just because a group contains many or few correct-answer digits.
Finish the Group Before Recording the Score
Sometimes a supplied stream ends in the middle of a trial. For example, a list may contain enough digits to complete one quiz and only seven digits of the next quiz. The seven digits do not make a complete 10-question trial. Continue generating three more digits for that same quiz, then count all ten digits together and record one score.
Worked Example: Continue an Incomplete Trial
Use 6 as the correct-answer digit. A stream begins with 2, 6, 1, 8, 6, 4, 0, 6, 3, 9, followed by 6, 5, 2, 6, 7, 1, 6. Simulate as many complete quizzes as possible. Then continue the stream with the digits 0, 6, 8.
The first ten digits form trial 1. There are three 6s, so trial 1 has a score of 3. The next seven digits begin trial 2: 6, 5, 2, 6, 7, 1, 6. At this point trial 2 is incomplete because only seven of its ten question outcomes are available. It already contains three 6s, but do not record a completed trial score yet.
Continue trial 2 with the next three digits, 0, 6, 8. The complete group of ten is 6, 5, 2, 6, 7, 1, 6, 0, 6, 8. It contains four 6s, so trial 2’s score is 4. The completed scores are therefore 3 and 4. The three extra digits complete trial 2; they do not start a new trial.
This boundary rule is the opposite of a stopping rule like the one in Simulating Repeated Trials Until Success. There, the outcome determines when a trial ends. Here, the number of questions determines when a trial ends, so every trial uses ten digits even if its score is known to be low or high before the last question.
What the Recorded Outcome Means
For each trial, the score is a count from 0 to 10. A score of 0 means no generated digit in that group was 6; a score of 10 means every digit in that group was 6. The scores between those extremes count how many of the ten guesses were correct. The order of the digits may differ, but only the count of 6s is recorded as the quiz score.
For instance, the sequences 6, 2, 1, 6, 3, 4, 5, 7, 8, 9 and 9, 8, 7, 6, 5, 4, 3, 2, 1, 6 both produce scores of 2. The correct answers occur in different question positions, but each sequence contains two correct-answer digits. Keep the full sequence long enough to identify the ten questions; then record its count.
Worked Example: Check an Extreme Score
A simulated quiz produces 7, 2, 0, 5, 8, 1, 4, 9, 3, 0. Using 6 as the correct-answer digit, what score should be recorded? What score would be recorded for the sequence 6, 6, 6, 6, 6, 6, 6, 6, 6, 6?
The first sequence contains no 6s, so the score is 0 correct out of 10. The second sequence contains ten 6s, so the score is 10 correct out of 10. Both are complete trials because each has exactly ten digits; neither requires a special stopping rule.
Common Mistakes and AP Exam Tips
- Stopping as soon as a correct answer appears. One 6 represents one correct question, not completion of a trial. Use all ten digits for the quiz.
- Generating a different number of digits for different scores. Every trial represents the same ten-question quiz, so every complete trial has exactly ten digits.
- Mixing up trial boundaries. Count the first ten digits together, then the next ten together. Do not restart a group whenever a 6 appears.
- Counting incorrect answers instead of correct answers. Under the stated assignment, count the 6s. The other nine digits represent incorrect answers.
- Changing the digit assignment during the simulation. Keep 6 as the correct-answer digit for every question and every trial.
- Recording a partial group as a score. If only seven digits of a trial are available, generate three more before recording the score.
- Describing the score as the number of correct trials. A trial is one quiz; its outcome is the number of correct answers on that quiz. State the count clearly, such as “4 correct answers out of 10.”
A full-credit simulation description identifies the chance model, explains what each digit represents, specifies the number of digits in a trial, and states exactly what is recorded. For this quiz, say that each digit represents one random guess, one specified digit represents a correct answer, ten consecutive digits make one trial, and the count of correct-answer digits is that trial’s score. If the random stream ends partway through a group, continue the same trial until ten digits have been generated.
Check Your Understanding
For each question, use 6 to represent a correct random guess and any other digit to represent an incorrect guess.
- A trial produces 6, 2, 6, 1, 8, 4, 0, 6, 3, 9. What score should be recorded?
- Why must a simulated quiz continue after the first correct answer appears?
- A stream contains ten digits for the first trial and seven digits for the second. How many additional digits are needed before the second trial’s score can be recorded?
- Two separate groups of ten digits each contain four 6s. What score does each trial receive, and are the trials still separate?
- What is the lowest possible recorded score, and what is the highest possible recorded score for one complete trial?