For this question, assume I have a standard deck of cards (52 cards, 13 of each set).
My question is: if I were to randomly draw 7 cards, how would I calculate the probability of at least one of those cards being a Spade?
Then, supposing no spades were drawn, how do I calculate the probability that the next card I draw is a spade?
For problem 1, it is easier to work out the probability that none are spades. This is $1 - 39/52 * 39/51 * 39/50 .... * 39/46) = .204$
the probability of at least 1 spade is 1 - the above, or about .8.
If you have drawn 7 cards and no spades, then there are 45 cards in the deck and 13 are spades, so the chance that the next is a spade is 13/45
There are $52 - 13 = 39$ non-spade cards in the deck, which means there are $39 \choose 7$ ways to draw 7 of them (ignoring order). You can divide by the total number of ways to draw 7 cards to get the probability of failing to draw a spade: ${{39 \choose 7} / {52 \choose 7}}$.
Subtracting from one to get the probability of success you get:
${1 - {{39 \choose 7} \over {52 \choose 7}}} = {535763 \over 605360} \approx 0.885032$.
The second part is simpler. If you already drew 7 cards, there are $52 - 7 = 45$ cards left. Of those, 13 are spades, so the odds of drawing a spade are $13/45 = 0.2\overline{8}$.
