Say I have a standard 52-card deck of cards. What is the probability of drawing an Ace of Spades? Presumably, $\frac{1}{52}$.

Say I have a standard 52-card deck of cards, and I know that the top card is an Eight of Clubs. What is the probability of drawing an Ace of Spades? This time, it's $\frac{0}{52}$, because I know what the top card is, and it's not an Ace of Spades.

How can exactly the same action have two different probabilities because of prior knowledge? I have done nothing to the deck in either case prior to drawing the top card, yet because I know what the top card is in the latter case, it affects the odds. Why?

(This almost sounds like a Schrödinger's Cat situation, but I can't tell if that's actually what's going on here or if I'm missing something.)

On a related note, I feel like drawing a card off the top of a deck is not a true random event. If I have a bag of 10 marbles, 3 of which are blue and 7 of which are red, what are the odds of drawing a red marble? Assuming I draw the marbles at random, the odds will always be $\frac{7}{10}$, regardless of how I place the marbles in the bag. But drawing a card is dependent on how I place the cards in the deck.

  • $\begingroup$ One is a conditional probability, so you're not computing the same probability in each case. $P(\text{ace is top card}) = \frac{1}{52}$ whereas $P(\text{ace is top card} | \text{top card is 8 of clubs}) = \frac{0}{52}$ $\endgroup$ – Brenton Apr 23 '17 at 19:27
  • $\begingroup$ What if you drew every card simultaneously? Then it's guaranteed you draw the Ace of Spades "next." $\endgroup$ – Dohleman Apr 23 '17 at 19:30
  • $\begingroup$ @Dohleman I mean picking the top card on the deck. $\endgroup$ – DonielF Apr 23 '17 at 19:35
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    $\begingroup$ If you like to think about the world in the deterministic way, then drawing a card is not a true random event. Nothing is truly random. Asigning probabilities to some events is just an expression of our uncertainty or the lack of knowledge. Hence "probability" is not a quality of the event itself, but also of your knowledge. $\endgroup$ – Michał Miśkiewicz Apr 23 '17 at 19:38

I think there's something implicit in the first statement that's important. When we talk about some event, like rolling a dice, we assume that dice is fair. When we talk about an unknown deck of cards, we assume the deck is randomly shuffled.

We say the probability of a random(ie, indistinguishable from any of the $52!$ possible decks) standard deck having the ace of spades on top is $\frac 1 {52}$ (as there are exactly $51!$ possible decks with this top card), while for any specific deck (fixed, static, and easily distinguished from all the other possible arrangements) the probability is either $1$ or $0$. We often assume, for ease of computation and sanity, that anything we don't want to investigate too closely is random with some distribution.

At the heart of things, there's also a philosophical question here. Are our abstract constructions (like probabilities or integers) fundamental to the universe, or consequences of our brains? This question is probably not going to be answered here, as we simply don't have one.

  • $\begingroup$ To be fair, die rolls are technically subject to chaos theory and are not completely random, but since we have no way of accurately measuring every single variable at play we treat it like random. But I like your analysis. $\endgroup$ – DonielF Apr 23 '17 at 21:41

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