Exercises: Interpretations of probability

State whether the following statements are true or false, justifying your answer:

1. a) "I have rolled this fair die $20$ consecutive times without getting a $5$. According to the law of large numbers, on the next roll it is very likely that a $5$ will finally come up."
2. b) "Even though no $5$ has come up in the last $20$ rolls, the probability of getting one on the next roll remains $\frac{1}{6}$: chance has no memory."

A drawing pin is dropped $400$ times and lands point up on $260$ occasions.

1. a) Estimate the probability that the drawing pin lands point up.
2. b) Explain why Laplace's rule cannot be applied in this experiment.
3. c) Which interpretation of probability did you use in part a)?

During a probability workshop in a Mathematics class, Cian flips a fair coin 8 consecutive times and gets the following sequence: H-H-H-H-H-H-H-H (eight heads in a row). Cian enthusiastically claims:

"On the ninth flip, the probability of getting tails is much greater than $0.5$, because the law of large numbers requires the frequency of heads and tails to balance out in the short term, so the coin must make up for the streak."

1. a) Identify and name the fallacy or reasoning error that Cian makes.
2. b) Formally explain why the probability of getting tails on the ninth flip is still exactly $0.5$, relying on the definition of independent events.
3. c) Clarify what the law of large numbers actually states: do past streaks get "compensated" in subsequent flips or are they "diluted" as the total number of flips increases? Justify your answer with clear reasoning.

Three 6th Year (Leaving Certificate) students are assigned three different situations to determine the probability of an event:

• Situation 1 (Aoife): Calculate the probability that, when drawing two cards at random from a 40-card Spanish deck, both are cups.

• Situation 2 (Seán): Determine the probability that a thumbtack, when tossed onto a flat table, lands point up.

• Situation 3 (Niamh): Estimate the probability that an Irish automotive company will go bankrupt during the next fiscal year.

1. a) For each situation, choose with justification which of the three interpretations of probability (classical Laplace, frequentist, or subjective) is appropriate.
2. b) Explain why in Situation 2 it is a serious mistake to apply Laplace's rule arguing that "there are only two possible outcomes: point up or point down, so the probability is $1/2$". What empirical procedure should Seán follow to obtain a reliable estimate?
3. c) In Situation 3, two financial analysts assign different probabilities ($0.05$ and $0.12$). Does this necessarily mean that one of them has made a calculation error? What essential mathematical condition must each analyst's assignment meet to be coherent?

The repeated tossing of a coin has been simulated on a computer and the relative frequency of HEADS has been recorded after $10$, $100$, $1,000$ and $10,000$ tosses, as shown in the graph.

1. a) Towards what value does the relative frequency seem to stabilise? What probability would you estimate for HEADS?
2. b) What law justifies this behaviour? State it in your own words.
3. c) Explain why the estimate with $10$ tosses is unreliable even if the experiment is correct.

At a fairground stall, a game called "the lucky disc" is played: a disc of $2\ \text{cm}$ in diameter is dropped onto a grid of squares of side length $5\ \text{cm}$, as shown in the figure, and you win if the disc lands without touching any line.

1. a) Calculate the theoretical probability of winning. (Hint: the position of the disc is determined by its centre; reason about which region of each square the centre must land in.)
2. b) A player drops the disc $300$ times and wins $114$ times. Calculate the relative frequency of winning and compare it with the theoretical probability.
3. c) Does the experimental result contradict the theoretical probability? Give reasons for your answer.

The weather app announces: "tomorrow, $70\ \%$ probability of rain".

1. a) Explain what this statement means: is it a classical (Laplace), frequentist or subjective probability? What data do you think it is based on?
2. b) Three friends assign probabilities to Saturday's match. The first says: "my team wins with probability $0.5$, draws with $0.3$ and loses with $0.3$". The second says: "it wins with $0.6$, draws with $0.3$ and loses with $0.1$". Are both assignments coherent? Why?
3. c) Can a subjective probability be "incorrect" even if it is coherent? Comment briefly.

A dart is thrown at a circular target made of three concentric circles of radii $r$, $2r$ and $3r$. Assuming that the dart always hits the target and that the probability of hitting a region is directly proportional to its area, calculate the probability that the dart hits the middle annulus (the region contained between the circle of radius $r$ and that of radius $2r$).

At a municipality's local festival, a fairground stall is designed with a square dartboard measuring $60\text{ cm} \times 60\text{ cm}$. Inside the dartboard, a circular band (a ring concentric with the center of the square) will be painted between two radii $r_1$ and $r_2$, with $0 < r_1 < r_2 \le 30\text{ cm}$. It is assumed that the impact of each dart is completely random and is modeled using a uniform distribution across the entire surface of the square.

1. a) If the inner radius is preset at $r_1 = 10\text{ cm}$, determine inversely the exact value of the outer radius $r_2$ so that the geometric probability of a dart landing in the prize band is exactly equal to $0.20$.
2. b) Imagine that the organizers decide to change the design and paint only a solid central circle of radius $R$. If the organizer claims: "To triple the probability of hitting the circle, it is enough to triple its radius $R$", reason algebraically whether this claim is true or false and calculate the exact factor by which the radius $R$ would have to be multiplied to triple the probability.

On an avenue in Galway, a pedestrian traffic light operates with a periodic cycle of 100 seconds divided into three continuous phases: 60 seconds of red, 5 seconds of flashing amber (warning of the end of the crossing time), and 35 seconds of green. A pedestrian arrives at the traffic light at a completely random and uniform instant $t$ within the cycle.

1. a) Before calculating, make a reasoned estimate: do you think the probability that the pedestrian must wait more than 30 seconds before the traffic light turns green is greater than, less than, or equal to $50\,\%$? Justify your prediction qualitatively based on the duration of the phases.
2. b) Mathematically model the problem using geometric probability on the interval $[0, 100]$ and calculate the exact probability that the pedestrian has to wait more than 30 seconds until the start of the green phase.
3. c) Check whether the result obtained confirms your initial estimate. Then, refute the following comment made by a passerby: "Since when arriving at the traffic light there are only two possible states for me (to wait or not to wait), the probability of waiting is $1/2$ by Laplace's rule."

A city council wants to estimate the proportion of residents who recycle daily. A survey of $2,000$ people chosen at random yields $1,340$ who claim to recycle daily.

1. a) Estimate the probability that a resident chosen at random recyclers daily and indicate which interpretation of probability you are using.
2. b) Another school repeats the survey with only $50$ people and obtains $31$ who recycle ($0.62$). Which of the two estimates is more reliable? What theoretical result justifies this?
3. c) Write a correct conclusion about the proportion of residents who recycle, mentioning the role of sample size.

Three people are discussing the probability that the 8:00 bus arrives late:

• Person 1: "Out of the last $200$ school days, it arrived late $24$ times, so the probability is $0.12$."

• Person 2: "The bus arrives either on time or late: two possible outcomes, so the probability of arriving late is $\frac{1}{2}$."

• Person 3: "I'd say there is a $40\ \%$ probability of it arriving late; that's what my experience tells me."

1. a) Identify which interpretation of probability (frequentist, classical or Laplace, and subjective) corresponds to each statement.
2. b) The reasoning of one of the people is incorrect. Which one is it and why?
3. c) What data would you collect to improve each of the three estimates?

After simulating $10,000$ coin flips on a computer, the relative frequency of HEADS turned out to be $0.5003$. Two students comment on the result:

• Student A: "The coin is biased, because if it were fair the relative frequency would be exactly $0.5$."

• Student B: "The exact probability of HEADS for this coin is $0.5003$."

1. a) Analyse the error in each of the two arguments.
2. b) Write down a correct conclusion about what the experiment allows us to state regarding the coin.

Two delivery lorries, $A$ and $B$, arrive at a loading bay independently and at random between 12:00 and 13:00 hours. Lorry $A$ needs 15 minutes to unload, whilst lorry $B$ needs 10 minutes. If a lorry arrives and the bay is occupied, it must wait. Using geometric probability (representing the arrival time of $A$ on the X-axis and that of $B$ on the Y-axis), calculate the probability that neither lorry has to wait for the other.

An agricultural cooperative in Wexford analyzes the springtime risk of suffering damage from two mutually exclusive meteorological phenomena on the same day: a late frost ($H$) or a premature heat wave ($C$). A Met Éireann historical series is available with the records of the last 200 equivalent spring days, in which there were 16 days with frost and 24 days with a heat wave.

Three organizations present reports with different proposals:

• Report 1: Assigns $P(H) = 16/200 = 0.08$ and $P(C) = 24/200 = 0.12$, justifying that these frequencies faithfully reflect the law of large numbers.

• Report 2: Argues that, since there has been no frost in the last three springs, the probability of frost for the next spring must rise to at least $0.25$, because "nature tends to compensate for anomalies in the short term to maintain the historical average."

• Report 3: Adjusts the probabilities based on dynamic climate projections and proposes a reasoned degree of belief: $P(H) = 0.05$, $P(C) = 0.35$, and a probability of mild weather (neither frost nor heat wave) equal to $0.60$.

1. a) Classify the interpretation of probability used in each of the three reports (Laplace, frequentist, subjective, or fallacious).
2. b) Refute the claim in Report 2 with mathematical arguments: explain the fundamental difference between "compensation" (gambler's fallacy) and "dilution" (law of large numbers) as the number of observed years tends to infinity.
3. c) Verify whether the assignment in Report 3 is consistent with Kolmogorov's axioms of probability. If so, calculate from it the probability that an extreme weather event is recorded ($H \cup C$) and justify whether it is legitimate to use subjective probabilities in decision-making when historical conditions are no longer stationary.

Two researchers, Aoife and Niamh, have arranged to meet at the laboratory of the Dublin Technology Park between 10:00 and 11:00 in the morning. Each will arrive independently and at random at some instant during that hour, with a uniform distribution. Aoife will wait at most 10 minutes for Niamh, while Niamh will wait for Aoife a maximum of $k$ minutes (where $0 < k \le 60$). Once their respective waiting time has elapsed, whoever has arrived will leave.

A student tries to solve the problem for the case $k = 10\text{ minutes}$ and reasons as follows:

"Each researcher can arrive at any of the 60 minutes of the hour, so there are $60 \times 60 = 3600$ possible pairs of minutes. For them to meet, the difference between their arrival minutes must be at most 10 minutes (that is, $-10, -9, \dots, 0, \dots, 9, 10$), which gives 21 possible differences. By Laplace's rule, the probability of meeting is $21/3600 \approx 0.0058$."

1. a) Explain in detail the two conceptual errors made by the student: one regarding the nature of the sample space (discrete vs. continuous) and another regarding the non-equiprobability of the arrival differences.
2. b) Model the problem using geometric probability in the region $[0, 60] \times [0, 60]$ of the Cartesian plane, where $x$ and $y$ represent the arrival minutes of Aoife and Niamh, respectively. State the inequalities that define the favorable region and calculate the exact probability of meeting when $k = 10\text{ minutes}$.
3. c) Inverse problem: If the management of the center requires that the probability of both researchers meeting is exactly $7/16 = 0.4375$, calculate the exact value of minutes $k$ that Niamh must commit to waiting for Aoife.