Two cards are drawn without replacement from an ordinary deck. Find the probability that the second is a spade, given that the first is the ace of spades.
To find the conditional probability in this case, notice that the first card has been remove from the deck so the total sample space for the second card has been reduced from the original 52-card deck. Notice that after the first card is drawn from the deck, there are 51 cards left to choose from.
This partial deck of 51 cards is the reduced sample space used to calculate the probability when the second card is drawn.
After the first card is drawn from the deck, there are 12 spades left in the deck.
Recall that if an event E is a subset of a sample space S, then the probability that event E occurs is P(E) = . Thus, the probability that the second card chosen is a spade is , or .
Therefore, the probability that the second card is a spade given that first is the ace of spades is .