Question

3. In Texas Hold 'Em, a "rainbow flop" occurs when the three cards in the flop all have different suits. Before any cards are dealt, what is the probability that the flop will be a rainbow? If you are dealt 13 cards from an ordinary, well-shuffled deck, what is the probability that all your cards have the same suit? What is the probability that all four aces appear in your hand? 4.

Answer 1

(3) The 1st card can be of any suit, the 2nd card has not to be of the same suit as of the 1st card, so we have now 39 cards left, again the 3rd card has not to be of the same suit as of the 1st and 2nd card, so we have 26 cards left.

1st card: doesnt matter, 2nd card: needs to be 39 out of 51; 3rd card: needs to be 26 out of remaining 50.

Thus, P (rainbow flop) = 1*39/51*26/50 = 0.3976

(4) If a single card is chosen than the probability that it is from a particular suit is 1/4.

So the success rate is 1/4 and the failure rate (it does not belongs to that particular suit) is 3/4.

So when 13 cards are chosen from a well shuffled deck then,

Probability that all cards will have same suit

=  ⁵²C_{13 (1/4)}¹³ (3/4)³⁹

^{If a single ace is chosen from a deck then its probability is 1/13.}

^{So the success rate is 1/13 and the failure rate(not an ace) is 12/13.}

^{So we have chosen 13 cards, so}

^{Probability of 4 aces in our hand is}

^{= 13}C₄ (1/13)⁴ (12/13)⁹