Question

Suppose a flop is said to "have a flush draw possibility" if it contains exactly two cards of the same suit, plus another card of a different suit. For instance, A♥ K♥ 10♠ would have a flush draw possibility, but A♥ K♦ 10♠ would not, and also A♥ K♥ 10♥ would not. What is the probability that the flop has a flush draw possibility?

Answer 1

A deck of 52 cards has 4 suits, Spades, Clubs, Diamonds and Hearts

There are 13 cards in each suit, from 2 till 10, the 3 face cards J, Q, K and finally Ace.

Therefore there are 4 cards of each type i.e 4 kings, 4 queens etc

Probability = Favourable Outcomes / Total Outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

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Total Outcomes= Choosing 3 cards out of 52 in ⁵²C₃ ways = 22100

First we choose a suit among the 4 from which we get 2 cards (of that suit) = ⁴C₁ = 4 ways

Then we choose 2 cards out of 13 from the chosen suit = ¹³C₂ = 78 ways

Then we choose a suit among the remaining 3 from which we get 1 cards (of that suit) = ³C₁ = 3 ways

Then we choose 1 card out of 13 from the chosen suit = ¹³C₁ = 13 ways

Therefore total favourable outcomes = 4 * 78 * 3 * 13 = 12168

Therefore the probability = 12168/22100 = 234 / 425 = 0.5506

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