AP Statistics, questions about probability with a deck of 52 cards

Question

We draw 5 cards out of a deck of 52 cards.

a. What is the probability that at least of those cards is Jack or a Queen

b. What is the probability that the first card picked from those 5 randomly chosen cards is a Clubs card (any Club card, 1-10, Jack, Queen, King)

c. If face cards are considered success and all others are considered failure, what is the probability of having at least one face card(success) within those 5 cards?

d. The probability of the second card from those 5 cards being a Spades card if the the first one was also a spades card.

e. The probability of having both a Jack or a Queen.

Answer 1

a) P(at least one is a jack or queen) = 1 - P(all are not jacks or queens)

As we are drawing 5 cards from 52, and 44 are not jack or queen), this equals

1 - C(44,5)/C(52,5) = 27017/46410 = .582137

b) P(first picked is a club) = 13/52 = 1/4 = .25

c) 16 of 52 cards are face cards, and 36 not face cards)

P(at least 1 face card) = 1 - P(no face cards) = 1 - C(36,5)/C(52,5) = 389/455 = .854945

d)P(first 2 cards are spades) = P(13,2)/P(52,2) = 13*12/(52*51) = 1/17 = .058824

e)P(both a jack and queen) = 1 - P(jack or not jack or queen) - P(queen or not jack or queen) + P(not jack and not queen) = 1 - (2 C(48,5) + 2 C(44,5))/C(52,5) = 6509/64974 = .100179