Exercises: Relative frequency and experimental probability

A die was rolled $50$ times and the $6$ came up $8$ times.

a) Calculate the relative frequency of the event "getting a $6$".

b) Compare it with the theoretical probability $\frac{1}{6}$. Does the result seem reasonable to you?

The arrow of a spinner with $4$ colours was spun $40$ times, yielding these results: red $12$ times, blue $10$, green $8$, and yellow $10$.

a) Make a table with the absolute and relative frequencies of each colour.

b) Which colour occurred with the highest relative frequency?

Tossing a coin $200$ times resulted in $98$ heads.

a) Calculate the relative frequency of the event "heads".

b) Would you suspect that the coin is biased? Why?

In a class of $30$ students, $12$ come to school by bicycle or scooter.

a) What is the relative frequency of this event in the class?

b) If the school has $600$ students, how many would you expect to come by bike or scooter? Is this value exact or only an estimate? Why?

Marcos rolls a conventional $6$-sided cubic die a total of $12$ times to check if it is loaded. He gets the number $5$ on $4$ occasions. After the experiment, Marcos states: "The relative frequency of rolling a $5$ is $4/12 = 1/3 \approx 0,33$. Since the theoretical probability is $1/6 \approx 0,17$, the die is clearly loaded. Furthermore, on the next roll it is almost impossible to get a $5$ again, because it has already come up too much."\n\na) Identify the two probabilistic reasoning errors Marcos makes and explain why they are incorrect.\n\nb) Why does a series of only $12$ rolls not allow you to be sure whether the die is loaded or not?\n\nc) If the die were fair and rolled $1\,200$ times, around what value would you expect the relative frequency of getting a $5$ to stabilize, and approximately how many times would that number come up?

At a charity fair booth there is an opaque urn with winning and non-winning tickets. After opening the first $50$ tickets of the afternoon, the relative frequency of winning tickets has been exactly $0,14$.\n\na) How many winning tickets have been drawn in those first $50$ draws?\n\nb) The organizers add new tickets and aim for the total relative frequency of prizes to rise to exactly $0,20$ by the end of the next $150$ draws (reaching a cumulative total of $200$ tickets played). How many winning tickets would necessarily have to be drawn in that second batch of $150$ draws?\n\nc) Explain why, even if the organizers designed the urn with that intention, the random nature of the draw does not make it $100\,\%$ certain that exactly that calculated amount will be drawn in the second batch.

A die is rolled an increasing number of times and the relative frequency of the event "getting a $4$" is recorded: after $10$ rolls it is $0,2$; after $50$, $0,18$; after $100$, $0,16$; after $500$, $0,164$; after $1\,000$, $0,167$.

a) What value does the relative frequency seem to settle at?

b) How is this value related to Laplace's rule?

c) How many "fours" would you expect in $6\,000$ rolls?

A trainer factory inspects $400$ pairs each day and records those with a defect: Monday $8$, Tuesday $5$, Wednesday $10$, Thursday $6$, and Friday $11$.

a) Calculate the relative frequency of defective pairs for each day.

b) Calculate the relative frequency of defective pairs for the whole week.

c) If a pair manufactured that week is chosen at random, what would you estimate the probability of it being defective to be?

The graph shows the relative frequency of "heads" when tossing a coin an increasing number of times.

a) Read the relative frequency off the graph after $80$ tosses.

b) What value does the relative frequency seem to settle at?

c) Do you think the coin is biased? Justify your answer.

A knucklebone (a sheep bone used as a die) has been tossed $600$ times and landed with the flat side facing up $264$ times.

a) What experimental probability would you assign to this outcome?

b) Explain why Laplace's rule cannot be applied in this experiment.

c) How many times would you expect it to land this way in the next $150$ tosses?

A teacher asks students to spin a spinner with three sectors $360$ times: red ($180º$), blue ($90º$), and green ($90º$). Three students submit these results:

a) Calculate the relative frequency of each colour in the three tables.

b) What would be the expected frequencies based on theoretical probabilities?

c) Two of the students have cheated. Who are they and why?

A jar lid is flipped $100$ times and lands with the top facing up $64$ times.

a) Estimate the probability that it lands face up.

b) And that it lands face down?

c) If it is flipped $500$ more times, how many times would you expect it to land face up?

d) Why can we not use Laplace's rule with the lid?

The graph shows the relative frequency of "heads" in $500$ tosses of a coin.

a) What value does the relative frequency settle at?

b) Is the coin biased? Why?

c) How many heads would you expect if it were tossed $1\,000$ more times?

A basketball player has an all-time free-throw percentage of $80\%$. In the last match, she scored only $3$ out of $5$. Her coach comments: "Her probability of scoring has decreased today."

a) Do you agree with the coach? Justify your answer by distinguishing between probability and frequency over a small number of trials.

b) What would you expect if she took $100$ free throws in upcoming training sessions?

During this month's training sessions, Lucía has taken $80$ free throws at the basket and made $68$.\n\na) Without doing written division or using a calculator, estimate with reasoning whether her relative shooting accuracy is closer to $70\,\%$, $85\,\%$, or $95\,\%$. Explain what you focus on mentally to make your estimate.\n\nb) Calculate the exact value of the relative frequency of made shots expressed as an irreducible fraction, a decimal number, and a percentage. Check whether your estimate from the previous section was correct.\n\nc) In the next game, Lucía takes her first $4$ free throws and misses all $4$. A spectator claims: "Lucía has lost her touch; her probability of scoring has plummeted." Relying on the relationship between sample size and the stability of relative frequencies, explain why the spectator's conclusion has no mathematical basis.

An AEMET weather station in a coastal town has recorded the daily weather during the month of October over the past $20$ years, accumulating a historical series of $620$ October days. In that record, noticeable precipitation was recorded on $217$ of those days.\n\na) Model the probability of it raining on any given October day in that town based on its historical relative frequency. Express the result as an irreducible fraction, a decimal number, and a percentage.\n\nb) The first $4$ days of October this year have been completely dry and clear. A neighbor states: "Tomorrow, day 5, it is definitely going to rain to make up for it, because the probability of rain is high." Reason critically whether this statement is valid from the standpoint of chance.\n\nc) If the month of October has $31$ days, how many rainy days would theoretically be expected throughout the whole month according to your model? Does this mean that in any given year that exact number of days will necessarily have rain? Justify your answer using the concept of sample variability.

We suspect that a die is biased and we roll it $600$ times. The $6$ comes up $168$ times.

a) Calculate the relative frequency of the event "getting a $6$".

b) What relative frequency would you expect if the die were fair?

c) Would you conclude that the die is biased? Why?

d) If we keep rolling it $300$ more times, how many "sixes" would you expect, based on your conclusion?

A supermarket has a bottle return machine that, on average, should not reject more than $2\%$ of valid containers. Over one week, the following data are recorded:

a) Calculate the relative frequency of rejection for each day, rounding to three decimal places.

b) Calculate the relative frequency of rejection for the whole week.

c) The manager must decide whether to call technical support. What do you advise? Justify your decision using the data.

After tossing a coin $4$ times and getting $3$ heads, Omar claims: "The probability of getting heads with this coin is $0,75$."

a) Explain the error in his claim.

b) What should Omar do to estimate the probability reliably?

c) If the coin is fair, what value will the relative frequency of heads approach as the number of tosses increases?

Two teams in a class experimentally estimate the probability of getting a total of $7$ when rolling two dice. Team A rolls $30$ times and gets this event $6$ times; Team B rolls $300$ times and gets it $50$ times.

a) Calculate the relative frequencies for both teams.

b) Calculate the theoretical probability of "total $7$" by applying Laplace's rule.

c) Which estimate is more reliable? Why?

In the 2nd year ESO mathematics lab, two challenges are proposed:\n- Challenge 1: Determine the probability that a thumbtack lands "point up" when tossed onto a table.\n- Challenge 2: Determine the probability of getting a sum equal to $7$ when rolling two standard cubic dice numbered from $1$ to $6$ simultaneously.\n\nTo solve them, two groups propose different methods:\n- Method A (Theoretical / Laplace's Rule): Write down the sample space, count the favorable and possible outcomes, and divide favorable outcomes by possible outcomes.\n- Method B (Experimental / Relative Frequencies): Toss the object $1\,000$ times, record how many times the event occurs, and calculate the ratio $f/N$.\n\na) For Challenge 1, choose which of the two methods is the only suitable one and rigorously justify why the other method would lead to a false result.\n\nb) For Challenge 2, analyze the validity of both methods. Which one would you choose as being exact and efficient without needing to carry out physical rolls? Justify your answer by explaining why Laplace's rule is applicable in this case.\n\nc) Write a general conclusion that serves as a guide for your classmates to know in which real-world situations it is strictly mandatory to resort to experimental probability and in which theoretical calculation is preferable.

The Dirección General de Tráfico (DGT) carries out preventive roadworthiness tests (ITV) on delivery vans. Data from two independent inspections in the same district are available:\n- Sample A (Quick check in an industrial estate): $50$ vans were inspected, and $8$ were found to have an expired ITV.\n- Sample B (Intensive district-wide campaign): $1\,200$ vans were inspected, and $132$ were found to have an expired ITV.\n\na) Calculate the relative frequency (as a percentage) of vans with an expired ITV in Sample A, in Sample B, and in the pooled set of both samples ($1\,250$ vans in total).\n\nb) A preliminary report claims that in the industrial estate (Sample A) the ITV problem is much more serious because the proportion obtained is $16\,\%$, whereas in the district (Sample B) it is only $11\,\%$. Using the law of large numbers and the influence of random fluctuations in small samples, refute that argument and explain why such a difference cannot be asserted.\n\nc) If $35\,000$ active delivery vans are registered in the provincial DGT database, which of the three relative frequencies from part a) provides the most reliable and rigorous estimate of the total number of vans circulating with an expired ITV? Justify your choice and calculate how many non-compliant vans would be estimated throughout the province using that value.