Six cards are drawn from a well-shuffled deck. You are told that at least an ace (A) is among the cards. What is the probability that there is at least a king (K) or at least a queen (Q)? (A deck consists of 52 cards, with 13 values (A; 2; 3; 4; 5; 6; 7; 8; 9; 10; J; Q;K) each having 4 suits }
I took cases in the denominator (1 ace, 2 aces, 3 aces...) and tried to take the complement (there are no kings, or there are no queens).
Let K: having at least a king
Q: having at least a queen
A: having at least an ace.
The probability of having at least a king or at least a queen given that there is at least an ace in the 6 cards drawn can be written as :
P(KUQ|A) = P(K|A) + P(Q|A) - P(K and Q | A)
We work to find P(A), and then use this relation for P(K|A):
P(K|A) = P(K and A) / P(A)
P(Q|A) is the same as P(K|A), and we use the same process and relations mentioned before to find P(K and Q|A).
Taking the complement was a big time-saver.