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.

• 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 $P(\text{ace is top card} | \text{top card is 8 of clubs}) = \frac{0}{52}$ – Brenton Apr 23 '17 at 19:27
• What if you drew every card simultaneously? Then it's guaranteed you draw the Ace of Spades "next." – Dohleman Apr 23 '17 at 19:30
• @Dohleman I mean picking the top card on the deck. – DonielF Apr 23 '17 at 19:35
• 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. – Michał Miśkiewicz Apr 23 '17 at 19:38

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.