Interpretations of Probability: Classical, Frequentist, and Subjective

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.

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 $0.7$."

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.

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 behavior? 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.

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.

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.

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.

At a high school, $400$ randomly chosen students were surveyed, and $268$ state that they play sports at least twice a week.

• a) Estimate the probability that a student at this school plays sports weekly.

• b) If the school has $1,200$ students, how many would you expect to play sports?

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

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 $\$16\,000$, calculate the "pure premium" (the price that would cover the risk exactly) and explain why the actual premium will be higher.