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(I also asked this question on Mathematics StackExchange. But since this question is more about the thinking and reasoning process, it is better to post it here, I think.)

These were very intuitive questions about probability. I did not give too much thought to them. However, my younger siblings asked me "why", and then I became confused. I came up with my own explanations. Please tell me whether my explanations are correct or not, and share with me your explanations so that I can teach my siblings and myself.

Question 1:

Assume the death rate of a disease is 35%, which means that 35 out of 100 patients die. Then mathematically speaking, without considering other conditions in real life, if somebody gets this disease, can we say the probability of him dying is 35%? Why can we think this way?

My explanation:

When we encounter this type of problem, we imagine we randomly think of ourselves as one of the 100 patients, 35 of who are going to die. Therefore, it is just like the chance of picking a red ball out of a box containing 65 white balls and 35 red balls.

Question 2:

Assume the school accepts 7 applicants per 100 applicants. Then mathematically speaking, without considering other conditions in real life, if somebody applies for this school, can we say the probability of him getting accepted is 7%? Why can we think this way?

My explanation:

When we encounter this type of problem, we imagine we randomly think of ourselves as one of the 100 applicants, 7 of which are going to be accepted. Therefore, it is just like the chance of picking a red ball out of a box containing 93 white balls and 7 red balls.

vincentlin
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  • What kind of questions are these? They are just telling you something and then asking you to repeat it. They don't seem like good questions. – causative May 27 '21 at 17:19
  • @causative Sorry for the questions. You might have misunderstood my intention. I provided my explanations but don't know whether they are acceptable or not, so I posted them here to get feedbacks. – vincentlin May 28 '21 at 06:56
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    You might find this answer relevant: 'Why is a measured true value “TRUE”?' https://philosophy.stackexchange.com/questions/81655/why-is-a-measured-true-value-true/81664#81664 Deaths are independent rather than like drawing lots, they are https://en.wikipedia.org/wiki/Poisson_distribution processes. Given all the data, an individual death is fully predictable. Probabilities relate to limits on our ability to distinguish between possible future scenarios (possible, based only on what we do know). Believers in karma, or theistic 'mysterious ways', might believe in hidden variables. – CriglCragl May 28 '21 at 21:59

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Your explanation is the most used frequentist probability interpretation of probability:

Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials. Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). This interpretation supports the statistical needs of many experimental scientists and pollsters. It does not support all needs, however; gamblers typically require estimates of the odds without experiments.

So the key here is you can repeat the experiment indefinitely like tossing a fair coin so eventually you'll approach a relative frequency of 0.5 for heads-up cases. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. If one cannot repeat an experiment, then another interpretation can be used called Propensity probability:

Theorists who adopt this interpretation think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind... Propensities are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. A central aspect of this explanation is the law of large numbers... This law suggests that stable long-run frequencies are a manifestation of invariant single-case probabilities.

A third major school of interpretation of probability is called Bayesian probability:

Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.

Double Knot
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  • Thank you for the comprehensive information about understanding probability. Do you think my explanations are acceptable if I want to teach my younger siblings about my Question 1 and 2? – vincentlin May 28 '21 at 07:09
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    @vincentlin In your frequentist explanation you need add a condition "without considering other conditions in real life" same as your question meaning principle of indifference is needed, like a fair dice each side having equal chance to be picked up, so under this (very unrealistic) assumption, imagining you repeat your picking experiment of random 35 people out of 100 for hundreds and thousands times or more, then you'll find the relative frequency (# of times when you picked as dead / total # of such experiments) will converge to 0.35, each such experiment is called simple random sampling. – Double Knot May 28 '21 at 17:11
  • Thank you for this answer. But I still don't know how to apply that relative frequency (0.35) to the case of a patient. If I, as a typical patient just like other individuals, get the disease, what is the chance of me dying? – vincentlin May 28 '21 at 17:31
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    @vincentlin you've already realized to ask your question you need to assume you're a *typical* patient first (since the reported a priori death rate is based on an ideal simplified assumption that each person having this disease is like atoms without any further difference, isolating other possible death causes for each case). Your frequentist interpretation of 35% chance to die conditioned on you having the disease rests on such indifference hypothesis. Of course in reality this may mean very little if other individual health conditions causally affect death additionally, like covid19... – Double Knot May 28 '21 at 18:07
  • Quantum probabilities seem to show the propensity interpretation, given the impossibility of local hidden variables. – CriglCragl May 28 '21 at 21:50