Exercises: Relative frequency and experimental probability

We flipped a coin $50$ times and heads came up $28$ times.

1. a) What is the absolute frequency of "heads"? And of "tails"?
2. b) Calculate the relative frequency of "heads" as a fraction and as a decimal.
3. c) And the relative frequency of "tails"?

A basketball player has scored $45$ of her last $60$ free throws taken.

1. a) What is her relative frequency of scoring?
2. b) If she takes a free throw tomorrow, what experimental probability would you assign to her scoring it? Express it as a fraction, as a decimal and as a percentage.

Cian is playing with a fair $1$-euro coin. He flips it $10$ consecutive times and gets $8$ tails and only $2$ heads. When he finishes, Cian states very confidently: “Since the theoretical probability of getting heads is $0.5$, in the next $10$ flips at least $8$ heads must necessarily come up to compensate for the streak and so that the relative frequency stabilizes at the theoretical probability.”

1. a) Identify the error in Cian's statement and explain why his reasoning is false by recalling the principle that “chance has no memory”.
2. b) Calculate the relative frequency of the event “getting heads” in Cian's first $10$ flips as a fraction and as a decimal. Does this result contradict the fact that the coin is fair?
3. c) If Cian were to continue flipping the same coin until completing $1,000$ flips, explain how the stabilization of the relative frequency actually takes place: does the coin need to “give back” the missing heads or does the impact of those initial $8$ tails simply dilute among the large total number of flips?

At a container recycling plant in Cork, an optical sensor inspects plastic bottles to separate the defective ones. During a morning test, the sensor detected exactly $18$ defective bottles, which accounted for a relative frequency of defects of $0.03$.

1. a) State the corresponding mathematical relationship and calculate how many bottles the sensor inspected in total during that morning test.
2. b) How many non-defective bottles passed through the sensor in that test? What was their relative frequency expressed as a percentage?
3. c) If in the afternoon the machine checks a new batch of $800$ bottles of the same type and under identical conditions, how many defective bottles is it expected to find? Justify your answer using the concept of experimental probability.

A group of students rolls a die $60$ times and records the results: $1$ comes up $8$ times; $2$, $11$ times; $3$, $9$ times; $4$, $10$ times; $5$, $13$ times; and $6$, $9$ times.

1. a) Construct the table of absolute and relative frequencies (the latter to three decimal places).
2. b) What "theoretical" relative frequency would you expect for each face if the die is fair? Express it as a fraction and as an approximate decimal.
3. c) Which outcome was furthest from expected? Do you think the die is rigged? Justify your answer taking into account the number of rolls.

You can do it on paper and send a photo with your answer.

In the playground, students play dropping a drawing pin: it can land point up or on its side. After $200$ drops, the drawing pin landed point up $112$ times.

1. a) Calculate the relative frequency of "point up" and of "on its side".
2. b) What experimental probability would you assign to each outcome?
3. c) Why can we not use Laplace's rule in this experiment to say that each outcome is equal to $\frac{1}{2}$?

A board game club organizes a raffle with a spinner of $5$ equal sectors numbered from $1$ to $5$. Before the draw, a spectator tests it by spinning it $500$ times and gets: $1$ comes up $97$ times; $2$, $103$ times; $3$, $98$ times; $4$, $101$ times; and $5$, $101$ times.

1. a) Calculate the relative frequencies of each number.
2. b) What theoretical probability does each number have if the spinner is fair?
3. c) Does the spinner seem balanced? Reason your answer.

In the science lab, two students are investigating the experimental probability that a metal soda bottle cap lands hollow side up or flat side up. After tossing the cap $250$ times on a smooth table under identical conditions, they observe that the flat side lands facing up on $165$ occasions.

1. a) Before doing any division with decimals, mentally estimate between which round percentage values the relative frequency of “flat side up” lies (for example, is it closer to $50\,\%$, $65\,\%$, or $80\,\%$?). Explain what quick mental reasoning you used to guide yourself.
2. b) Check your estimate by calculating the exact relative frequency of “flat side up” and “hollow side up”, expressing them as a decimal number and as a percentage to one decimal place.
3. c) If a classmate is going to toss that same cap only $5$ times in a quick game, is it certain that it will land flat side up in exactly the proportion calculated in the previous section? Justify why experimental probability only offers reliable guarantees when the number of repetitions is large.

In a Mathematics class, students want to find out the probability that a solid wooden pyramid with a square base and four identical triangular faces lands resting on its square base when tossed into the air. To investigate this, two proposals arise:

• Aoife's method: Apply Laplace's rule directly, reasoning that, since the solid has $5$ faces in total and one of them is the base, the theoretical probability is $\dfrac{1}{5} = 0.2$ ($20\,\%$).

• Seán's method: Toss the pyramid $400$ times onto a cushioned surface, record how many times it lands resting on the base, and calculate the relative frequency of that event.

1. a) Compare both methods and justify which one is suitable for this experiment. What essential condition does Laplace's rule require that is not met in this geometric object?
2. b) When applying the correct method and tossing the pyramid $400$ times, the base lands resting on $144$ occasions. Assign the experimental probability to the events “resting on the base” and “resting on a triangular face”, expressing them as irreducible fractions, decimals, and percentages.
3. c) If a third student suggested tossing the pyramid only $10$ times to finish sooner, explain why their estimate would not be scientifically reliable compared to the one with $400$ tosses.

The graph shows how the relative frequency of the event "getting heads" changes as a coin is tossed more and more times.

1. a) Which value does the relative frequency approach as the number of tosses becomes very large?
2. b) Why do you think the fluctuations are large at the beginning and small at the end?
3. c) If we flipped the coin $100,000$ times, what relative frequency would you expect to get, approximately? And how many heads?
4. d) Does this graph contradict the statement "in $10$ tosses, $5$ heads must come up"? Reason your answer.

Two basketball players boast about their accuracy:

• Lara has scored $12$ of her last $20$ free throws.

• Miriam has scored $45$ of her last $100$ free throws.

1. a) Calculate the relative frequency of scoring for each player.
2. b) Which of the two has a better success rate?
3. c) The coach has to choose one of them to take the decisive shot and says: "I trust Miriam's data more because it comes from many more shots." What does he mean? Do you agree with his reasoning? Argue using what you know about relative frequency.

A mathematics app simulates rolling a die $11,000$ times and obtains these results: $1$ comes up $1,660$ times; $2$, $1,668$ times; $3$, $1,655$ times; $4$, $1,670$ times; $5$, $1,647$ times; and $6$ comes up $2,700$ times.

1. a) Calculate the relative frequency of each outcome, rounded to four decimal places.
2. b) If the virtual die were fair, what relative frequency would you expect for each face, approximately?
3. c) Which face do you suspect is "loaded"? Justify your answer by comparing the obtained frequencies with the expected ones.

Met Éireann has the historical record of the last $50$ years of precipitation in the city of Cork during the month of May (which has $31$ days). In that historical period of $1,550$ observed days of May, precipitation was recorded on $310$ of them.

1. a) Model this situation by calculating the experimental probability that it rains on any given day of May in Cork, expressed as an irreducible fraction, as a decimal number, and as a percentage.
2. b) In the current year, during the first $10$ days of May it rained on $5$ of them. Calculate the relative frequency of rain in that first ten-day period. Does this figure allow us to state that Met Éireann's historical model is incorrect? Support your answer by analyzing the difference between short-term random behavior and long-term stabilization of frequencies.
3. c) If during the remaining $21$ days of that month of May it did not rain on any day, what would be the relative frequency of rainy days for the entire month of May of this year? Compare this final value with the historical probability and explain what probabilistic principle this change illustrates compared to the data from the first $10$ days.

Two groups of students want to check whether an old coin is biased in favor of heads or if it is a fair coin. To study this, they have $100$ flips already performed, divided into $5$ consecutive rounds of $20$ flips each, with the following counts of “heads”:

• Round 1: $13$ heads out of $20$

• Round 2: $8$ heads out of $20$

• Round 3: $14$ heads out of $20$

• Round 4: $9$ heads out of $20$

• Round 5: $12$ heads out of $20$

Group A analyzes the rounds separately and concludes: “In $3$ of the $5$ rounds, the relative frequency exceeded $0.55$ ($65\,\%$, $70\,\%$, and $60\,\%$). Since most of the rounds favor heads, it is proven that the coin is loaded.”

Group B argues that this analysis is flawed and proposes combining all the flips and extending the experiment to reach at least $1,000$ flips before drawing conclusions.

1. a) Point out Group A's methodological error. Why can classifying short rounds in isolation lead to a false conclusion in a random phenomenon?
2. b) Calculate the overall relative frequency of the event “getting heads” across the set of $100$ flips performed (in decimal form and as a percentage). If the coin were fair, is this result of $100$ flips conclusive proof that it is biased, or can it be attributed to the natural fluctuation of chance? Justify your answer.
3. c) Justify why Group B's proposal to expand to $1,000$ flips is superior: describe what mathematical pattern the cumulative relative frequency would show if the coin were well-balanced compared to the pattern it would show if it were actually biased with a theoretical probability of heads equal to $0.60$.

Topic challenge. Can we predict the weather with mathematics? Become a weather analyst with this challenge.

1. Sample space and data: Go to the Met Éireann website and record in a spreadsheet whether it rained or not in your town over the last $30$ days. Define the sample space $E$ for the daily weather.

2. Laplace versus reality: Calculate the relative frequency of the event "rainy day" ($f$). Consider whether it makes sense to apply Laplace's rule by assigning $P = 1/2 = 0.5$ to it raining. Justify whether both outcomes are equally likely in your municipality.

3. Experimental simulation: Flip a coin $50$ times (heads = rain, tails = no rain) to simulate an equally likely model. Record the cumulative relative frequencies every $10$ flips and observe their tendency toward the theoretical probability.

4. Final product: Prepare a brief report (one page) with a comparative chart between your flips and the actual Met Éireann records, explaining the differences between experimental and theoretical probability.

Assessment criteria:

• Correct identification of the sample space and rigorous justification regarding the use of Laplace's rule.

• Proper use of the spreadsheet to tabulate data and calculate relative frequencies with precision.

• Thoughtful comparison between the random experiment and real climate data.

• Clarity of presentation in the conclusions of the final report.