A workbook is made up of 3 chapters as described below.

Question

Chapter 1 is 45 pages long;
Chapter 2 is 60 pages long;
Chapter 3 is 55 pages long.
Dominic opens the workbook to a random page and records the chapter number. He repeats this process 50 times. The results are shown in the table below.

In the first trial, the spinner landed on A is 2, on B is 3, on C is 5. In the second trial, the spinner landed on A is 2, on B is 2, on C is 6. In the third trial, the spinner landed on A is 3, on B is 3, on C is 4.

Which of these statements are true? Choose all that are correct.

A.
The observed relative frequency of opening the book to a page in chapter 1 is larger than the theoretical probability.

B.
Of Dominic's trials, 20% resulted in opening the book to a page in chapter 3.

C.
If Dominic repeated his experiment 160 times, then he would open the book to a page in chapter 1 exactly 45 times.

D.
The theoretical probability that the book will be opened to a page in chapter 2 is 0.375.

E.
Dominic made a mistake in his experiment because the observed relative frequency for opening the book to a page in chapter 2 is 50%.

Answer 1

To assess which statements are true based on the scenario with Dominic's trials and the theoretical probabilities of the chapters, let's first establish the total number of pages in the workbook and the probabilities of landing on each chapter.

Total pages in the workbook:

• Chapter 1: 45 pages
• Chapter 2: 60 pages
• Chapter 3: 55 pages

Total pages = 45 + 60 + 55 = 160 pages.

Theoretical probabilities:

• Probability of landing on Chapter 1 = \( \frac{45}{160} = 0.28125 \)
• Probability of landing on Chapter 2 = \( \frac{60}{160} = 0.375 \)
• Probability of landing on Chapter 3 = \( \frac{55}{160} = 0.34375 \)

Next, let's examine the results from Dominic's trials:

Trial results:

1. A: 2, B: 3, C: 5
2. A: 2, B: 2, C: 6
3. A: 3, B: 3, C: 4

Now, let’s summarize the total counts per chapter from these results:

• Total counts for Chapter 1 (A):
  • Trial 1: 2
  • Trial 2: 2
  • Trial 3: 3
  • Total for Chapter 1 = 2 + 2 + 3 = 7
• Total counts for Chapter 2 (B):
  • Trial 1: 3
  • Trial 2: 2
  • Trial 3: 3
  • Total for Chapter 2 = 3 + 2 + 3 = 8
• Total counts for Chapter 3 (C):
  • Trial 1: 5
  • Trial 2: 6
  • Trial 3: 4
  • Total for Chapter 3 = 5 + 6 + 4 = 15

Number of trials = 3 (as we only have the results of 3 trials).

Calculating the observed relative frequencies:

• Relative frequency of Chapter 1: \( \frac{7}{3} \) (not practical since it exceeds 1, the issue might arise from errors in trials)
• Relative frequency of Chapter 2: \( \frac{8}{3} \)
• Relative frequency of Chapter 3: \( \frac{15}{3} \)

Now we’ll evaluate each statement:

A. The observed relative frequency of opening the book to a page in chapter 1 is larger than the theoretical probability.

• Chapter 1’s theoretical probability is \( \approx 0.28125 \), observed result is \( \frac{7}{3} \). This statement is true if calculations were based on proper counts.

B. Of Dominic's trials, 20% resulted in opening the book to a page in chapter 3.

• The relative frequency for Chapter 3 is \( \frac{15}{3} \) = 5, which is not a percentage point but a frequency. This statement is false.

C. If Dominic repeated his experiment 160 times, then he would open the book to a page in chapter 1 exactly 45 times.

• Since the \( 160 \) trials are theoretical, that statement isn’t practically realistic to expect exact counts. The probability was approximately 28.125%. This statement can be determined false.

D. The theoretical probability that the book will be opened to a page in chapter 2 is 0.375.

• This is true as derived from previous calculations.

E. Dominic made a mistake in his experiment because the observed relative frequency for opening the book to a page in chapter 2 is 50%.

• This is false, an incorrect supposition per the observed results because we can't derive that count over this infinitude based on just three trials.

Conclusions: Truthful Statements are: A and D.