Suppose we have six different coloured decks of cards numbered 1 to 10, ordered ascendingly with 1 on top. Now we're going to draw cards, no replacement. The choice of deck is independent (let's say a die decides). This leads me to believe that, in any given round, the probability of drawing the top card is $1/|decks|$, so $1/6$ until the first deck is emptied, $1/5$ from then until the second deck is emptied and so on. Is this correct?
Now, the probability of drawing, say, four cards from the same deck out of four draws, is the probability of drawing the first card followed by the second card and so on. But what if, instead, precisely one of those cards was from any other stack?
- not blue
First let's say we don't care when in the sequence the different coloured card is drawn. What is the probability of this outcome? The probability of drawing three same-coloured cards in sequence plus the probability of not drawing that colour? That doesn't sound right to me but with mutual exclusivity I'm not sure how to limit the outcome as specified. Does this throw conditional probability out the window?
Now we'll assume that the alien card was neither the first nor the last card; or more generally, we are given a set of draws that may have resulted in a different colour and another set that definitely did not. The constraints are otherwise the same. How does this change the game?