"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.

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.

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.

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."

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 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.

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.