Question 1: A bag contains 60 red balls and 40 blue balls. If a ball is randomly drawn and replaced 50 times, and red balls are drawn 28 times, what is the experimental relative frequency of drawing a red ball?

Answer: 0.56

Question 2: Calculate the relative frequency of rolling a 4 on a fair six-sided die after 120 rolls in which a 4 appeared 25 times.

Answer: 0.2083

Question 3: A survey reports that out of 200 students, 120 have used the school library in the past month. What is the relative frequency of students who have not used the library?

Answer: 0.4

Question 4: In an experiment, a coin is flipped 100 times, and heads appears 47 times. What is the relative frequency of tails?

Answer: 0.53

Question 5: A student claims that the relative frequency of drawing an even number on a die is 0.5. In 60 rolls, they observed 32 even numbers. Is their claim accurate? Explain your reasoning.

Answer: Relative frequency = 32/60 ≈ 0.533. The claim is approximately correct, but slightly underestimated.

Question 6: A biased spinner has sections labeled 1, 2, 3, 4 with probabilities 0.1, 0.2, 0.3, 0.4 respectively. After 100 spins, the outcomes are: 1 (8 times), 2 (22 times), 3 (30 times), 4 (40 times). Calculate the relative frequencies and compare with the theoretical probabilities. Comment on any discrepancies.

Answer: Observed: 1=0.08, 2=0.22, 3=0.30, 4=0.40. Discrepancies are small; relative frequencies roughly match expected probabilities, indicating experimental data aligns well with theory.

Question 7: Identify and correct the mistake: A student says that the relative frequency of drawing a card from a deck and getting a heart is 13/52 = 0.25, but then they say that the experimental relative frequency from 100 draws was 0.3. Are these calculations consistent? Why or why not?

Answer: The theoretical relative frequency is 13/52 ≈ 0.25, but experimental data (0.3) may differ due to chance. The mistake is assuming the theoretical probability equals the experimental relative frequency; they are expected to vary.

Question 8: A die is rolled 1000 times. The observed frequency of landing on 6 is 180. What is the relative frequency, and how does it compare to the theoretical probability?

Answer: 0.18; slightly higher than the theoretical 1/6 ≈ 0.1667, but within expected variation.

Question 9: The teacher shows students 50 data points for the number of times a coin lands heads in 100 flips, with the mean being 52. The teacher claims the relative frequency is exactly 0.52. Is this correct? Explain.

Answer: The relative frequency based on data is 52/100 = 0.52, so the teacher's claim is correct. Minor random variations are expected in experimental data.

Question 10: Challenge: After 250 coin flips, the relative frequency of heads is 0.49. What can you infer about the fairness of the coin? Explain your reasoning.

Answer: A relative frequency of 0.49 is very close to 0.5, suggesting the coin is likely fair. Small deviations are due to randomness.

Question 11: Identify the common mistake: A student calculates the relative frequency of drawing a blue ball as 15/50 = 0.3, but then concludes this means the probability is exactly 0.3. Is this correct? Why or why not?

Answer: Incorrect. Relative frequency is an estimate based on data, not the exact probability. The true probability might differ; more data reduces this error.