Mechanics of and intuition behind probabiliity

Question

Suppose in a given experiment that an event $E$ will occur with probability $p$; if the experiment is staged only once, then we know from the previous statement that the probability that the event $E$ occurs is given by $p$.

Now, say we stage the experiment $n$ times (where $n \to \infty$ and experiments are independent); intuitively (and mathematically, too, I guess), because the probability is constant across "stagings" of the experiment, I have deduced (perhaps erroneously) that the probability (before any staging has taken place) of the event $E$ occurring once across $n$ stagings is also given by $p$.

However, this seems to conflict with the frequentist definition of probability, paraphrased by the following:

Given $p = \frac{x}{y}$, as $n \to \infty$, it should be expected that, on average, event $E$ will occur $x$ times per every $y$ iterations of the experiment.

Consider an example:

A man with a deck of cards (52 in total, with 13 of each suit) provides you with the following (unrealistic) proposition:

You have $n$ chances (i.e. $n$ draws with replacement and reshuffling, where $n \to\ \infty$) to randomly select any card from the the suit of spades. If you're successful, you acquire the man's life savings; however, if you're unsuccessful, he acquires yours.*

*(In order to eliminate complex scenarios, assume both life savings are equal and that you are risk neutral (i.e. indifferent to risk).)

The first interpretation above suggests that the probability of selecting any card from the suit of spades across $n$ draws remains at $p = \frac{13}{52} = \frac{1}{4}$; this implies that the chance if this event occurring is significantly less than the chance of this event not occurring ($\frac{1}{4} < \frac{3}{4}$); hence you should not take this bet.

The second interpretation suggets that because $n \to \infty$, we should expect the event to occur, on average, once every $4$ times; you have $n$ chances (where $n \to \infty \implies n >> 4$). We should take this bet as the event will (almost?) definitely happen many times across $n$ draws.

It should also be noted that the first interpretation suggests that the event may never occur (in fact, it is more likely that it won't occur!), whilst the second interpretation suggests that as $n \to \infty$, the event will occur infinitely many times!

I suppose my first question should be: In the scenario above (replacement, independence, etc.), is my reasoning that the probability $p$ of event $E$ occurring across $n$ stagings constant, and, hence, equivalent the probability of the event occurring across one staging of the experiment correct?

Secondly, assuming the answer to the previous question is affirmative, is the perceived conflict presented in the above scenario founded on sound principles? If so, how should we progress in the presence of this conflict?

Answer 1

The intuition you express in your second paragraph

Now, say we stage the experiment n times (where n→∞ and experiments are independent); intuitively (and mathematically, too, I guess), because the probability is constant across "stagings" of the experiment, I have deduced (perhaps erroneously) that the probability (before any staging has taken place) of the event E occurring across n stagings is also given by p.

is, indeed, erroneous. That is the probability of the event occurring on any one staging. The probability of it occurring at least once does tend to 1. For probability p with k stagings, the probability is $(1-p)^k$.

Answer 2

You won't go very far with this typical textbook interpretation of probability, because it is circular: if the probability of an event is a real number that is likely to be close to the fraction of occurrences of the event in a very long series of repetitions of some experiment, how can we possibly interpret what likely means without resorting to some previous definition of probability? You have two ways to explore this question: 1) Look for interpretations of probability based on concepts such as algorithmic complexity; in this case, be prepared to deal with very subtle problems in the Foundations of Mathematics; you probably can't enter this field of research without a solid background in Mathematical Logic; 2) Do it the Bayesian way, either with a construction like the one in DeGroot's Optimal Statistical Decision, or with a Dutch Book argument, as described in Schervish's Theory of Statistics. This other question may be helpful.