Exercises: Interpretations of probability: classical, relative frequency and subjective

A die has been rolled $200$ times and these results were obtained:

• a) Calculate the relative frequency of each face.

• b) Estimate the probability of rolling an even number with this die.

• c) Do your estimates match the probabilities given by Laplace's rule? Comment on the difference.

True or false, giving reasons for your answer: "I have tossed a coin $10$ times and obtained $7$ heads; therefore, the probability of getting heads on this coin is no longer $0,5$, but $0,7$."

Assign a subjective probability, between $0$ and $1$, to each of the following events and justify your choice:

• a) "Tomorrow it will rain in your city."

• b) "You will pass the next mathematics exam."

• c) Explain why in these cases you cannot use either Laplace's rule or relative frequencies.

On a European roulette wheel ($37$ pockets: $18$ red, $18$ black and the green zero), $5.000$ spins have been recorded, and the ball landed on red $2.560$ times.

• a) What probability of "red" does the classical interpretation (Laplace's rule) assign?

• b) What probability does the relative frequency interpretation assign based on the data?

• c) Comment on which of the two you would give more credibility to, and why.

Marcos and his classmate Elena are studying the probability of different random experiments. When analyzing the toss of a thumbtack onto a table, Marcos claims:

"Upon landing, the thumbtack can only end up in two positions: point up or resting on its side with the point down. Since there are two possible outcomes and one favorable outcome, the probability that it lands point up is exactly:

applying Laplace's rule."

• a) Explain in detail why Marcos's reasoning is flawed and which essential requirement of Laplace's rule is not met in this case.

• b) Elena suggests using the frequentist interpretation. Describe what experiment they would need to carry out, what data they should record, and how they would calculate the probability from them.

• c) After tossing the thumbtack $400$ times under the same conditions, they observe that the point ends up facing up on $136$ occasions. Calculate the relative frequency obtained, express it in decimal form, and justify whether it is reasonable to adopt this value as an estimate of the desired probability.

A logistics and van delivery company in the Seville metropolitan area wants to know the probability that one of its vehicles will suffer a breakdown during a workday. The management requests advice, and two methodological alternatives are presented:

• Method A: Consider that a van can only end the day in two ways (with a breakdown or without a breakdown) and apply Laplace's rule by assigning $P(\text{breakdown}) = 1/2 = 0,5$.

• Method B: Review the fleet's IT records from the past year, which show $15.000$ completed workdays and a total of $180$ breakdown incidents, to calculate the corresponding relative frequency.

• a) Compare both methods and justify which of the two is appropriate for this problem, pointing out the fundamental flaw of the discarded method.

• b) Calculate the estimated probability of a breakdown per workday using the correct method and express it in decimal form and as a percentage.

• c) If the company expects to carry out $3.500$ delivery workdays next quarter, how many breakdowns can be expected to occur? Explain which mathematical principle supports the reliability of this estimate.

The graph shows the evolution of the relative frequency of the event "heads" when repeatedly tossing a coin, as shown in the figure.

• a) Describe what happens to the oscillations of the relative frequency as the number of tosses increases.

• b) What law states this behaviour? Write it in your own words.

• c) What would you expect the relative frequency to be if $10.000$ tosses are reached?

A board game die is suspected to be biased. It is rolled $1.000$ times and the following results are obtained:

• a) Estimate the probability of each face using relative frequencies.

• b) Calculate the estimated probabilities of "even", "less than $6$" and "rolling $1$ or $2$".

• c) In view of the estimates, does the die appear to be fair? Justify your answer.

A simulation app virtually rolls two fair dice $5.000$ times. In $833$ of the simulations the sum of the dice was $7$.

1. a) Estimate the probability that the sum is $7$ from the simulation.

1. b) Calculate the exact theoretical probability of that event using Laplace's rule.

1. c) Explain what causes the small difference between the two values.

At a secondary school, $400$ randomly chosen students were surveyed, and $268$ state that they practise sport at least twice a week.

• a) Estimate the probability that a student at this school practises sport weekly.

• b) If the school has $1.200$ students, how many would you expect to practise sport?

• c) Explain which interpretation of probability you used and why the classical interpretation is not applicable here.

A casino introduces a new electronic French roulette wheel that has $37$ pockets numbered from $0$ to $36$. To verify that the random number generator does not present anomalies or biases, an auditor examines the behavior of the event $A = \text{“getting the number } 0\text{”}$.

• a) Before analyzing any empirical data, calculate the theoretical probability of $A$ using Laplace's rule (rounding to four decimal places) and estimate the approximate percentage of spins in which $0$ should appear if the device operates with perfect equiprobability.

• b) In an initial short test of $N = 80$ spins, the number $0$ appears $5$ times. Calculate the relative frequency observed in this series. Can it be concluded from this result that the machine is rigged in favor of $0$? Justify your answer using the law of large numbers.

• c) The audit is expanded to record $N = 10.000$ consecutive spins, obtaining the number $0$ on $274$ occasions. Calculate the new relative frequency, compare it with the theoretical estimate obtained in section a), and argue whether you now have sufficient basis to validate the software's operation.

An engineering team is evaluating the reliability of the automatic emergency braking system of an electric car prototype. After subjecting the system to a continuous series of identical track tests, the estimated frequentist probability of correct braking obtained is $P(\text{success}) = 0,985$.

• a) Throughout the testing phase, the system failed on exactly $24$ occasions. Set up an equation relating the observed relative frequency of failure to the total number of tests $N$ performed, and solve it to determine how many tests were conducted in total.

• b) Design the contents of a hypothetical urn containing only red balls (representing failure) and green balls (representing success), such that the classical probability of randomly drawing a red ball exactly matches the frequentist probability of failure found in the tests. Specify the minimum possible number of balls of each color that the urn must contain.

• c) After observing a streak of $120$ consecutive tests without a single failure, a team member proposes canceling the remaining tests, claiming: "Since the failure rate is $1,5\,\%$, the system has already accumulated too many consecutive successes, and the probability of it failing on the next test is imminently high to balance out the average." Identify the name of this reasoning error and refute it using the properties of probability.

An insurance company sets the price of bicycle theft insurance based on its records: out of $20.000$ bicycles insured last year, $340$ were stolen.

• a) Which interpretation of probability does the insurance company use? Reason why the other two interpretations (classical and subjective) would not be suitable.

• b) Calculate the estimated probability of theft.

• c) If a bicycle is worth $800\ \text{€}$, calculate the "pure premium" (the price that would cover the risk exactly) and explain why the actual premium will be higher.

State, with reasons, whether each of the following statements about a fair die is true or false:

• a) "I have gone $20$ rolls without rolling a $5$; by the law of large numbers, on the next roll it is very likely that a $5$ will appear."

• b) "Even though no $5$ has appeared in $20$ rolls, the probability of rolling $5$ on the next roll is still $\frac{1}{6}$."

• c) "If I roll the die $6.000$ times, exactly $1.000$ fives will come up."

The figure shows two simulations of a random experiment whose event of interest has probability $0,3$: panel A with $100$ repetitions and panel B with $10.000$, as shown in the figure.

• a) Explain why the final relative frequency in panel A deviates more from $0,3$ than that in panel B.

• b) State the law of large numbers precisely.

• c) Does this law guarantee that the relative frequency will equal exactly $0,3$ at some point? Give reasons for your answer.

We want to estimate experimentally the probability that, when dealing $4$ cards from a Spanish deck of $40$ cards, there is at least one ace, using a spreadsheet with its random number generator.

• a) Describe the simulation procedure in detail: how to code the cards, how to simulate dealing without replacement and how many repetitions you would perform.

• b) Explain how you would obtain the estimate and what you would do to check that your simulation is reliable.

The National Statistics Institute (INE) periodically publishes mortality tables for the population in Spain. According to the data for a cohort of $65$-year-old individuals (the typical retirement age), out of every $100.000$ people who reach that age, $88.500$ survive to age $75$ and $63.720$ reach age $85$.

• a) Mathematically model the following two probabilities from these demographic data: the probability $P_1$ that a randomly chosen $65$-year-old person dies before turning $75$, and the conditional probability $P_2$ that a person who has already turned $75$ lives to celebrate their $85$th birthday. Express the results as an irreducible fraction and to three decimal places.

• b) Reason in detail which interpretation of probability (classical, frequentist, or subjective) underpins the calculations in the previous section. Explain why it would be a serious methodological error to apply Laplace's rule by arguing that for each year a person has only two options: "remain alive or die".

• c) A financial institution offers a life annuity pension plan to an individual $65$-year-old client. Although the institution takes the INE tables as an initial baseline, an actuary adjusts the probability after assessing the client's medical reports, lifestyle habits (smoking, exercise), and family history. Explain the epistemological transition that occurs in this process: what role does frequentist probability play as a starting point, and how does it transform into a subjective (or Bayesian) probability? Why would neither interpretation alone be sufficient to manage the insurance business?

Two Social Sciences students, Lucía and Tomás, are analyzing the forecast that AEMET (State Meteorological Agency) has published for next Saturday in their municipality: "Probability of precipitation: $80\,\%$, with a yellow warning for storms."

Lucía says:

"That $80\,\%$ is a purely subjective judgment of the meteorologists, without objective mathematical rigor. In reality, for Saturday there are only two logical possibilities: either it rains or it doesn't rain. Since we cannot predict the future with certainty, applying Laplace's rule, the true probability is $1/2 = 0,5$."

Tomás counters:

"It is not subjective; it is strictly frequentist: it means that during $80\,\%$ of the hours of the day it will be raining over the municipality, that is, about $19$ hours and twelve minutes of continuous rain. Moreover, since it has rained non-stop for the past four Saturdays in a row, by the law of large numbers the actual probability of it not raining this Saturday must be well above $20\,\%$ to compensate for the monthly historical average."

• a) Refute Lucía's stance: identify and rigorously argue the two conceptual errors she makes (one regarding the conditions for applying Laplace's rule and another regarding the scientific nature of meteorological models).

• b) Refute Tomás's stance: identify and rigorously argue his two errors (one regarding the physical interpretation of the precipitation percentage and another regarding how the law of large numbers works and the gambler's fallacy).

• c) Explain how that $80\,\%$ is calculated in modern meteorology using ensemble forecasting models (running dozens of numerical simulations with slight perturbations in the initial conditions) and discuss with justification which of the three interpretations (classical, frequentist, or subjective/degree of belief based on evidence) this approach falls under.