Question

Suppose we are playing draw poker. we are dealt (from a well-shuffled deck) five cards, which contain four spades and another card of a different suit. we decide to discard the card of a different suit and draw one card from the remaining cards to complete a flush in spades (all five cards spades). determine the probability of completing the flush.

Answer 1

Assuming a standard deck, we started with 52 prior to dealing, then drew 4 spades and 1 non-spade. This leaves 47 cards in the deck. From these 47, we hope to draw another spade. The deck originally started with 13 spades, and after dealing the five cards, the remaining deck contains 9 spades. The probability of drawing one of these 9 cards from the deck of 47 is

$\dfrac{\binom91\binom{38}0}{\binom{47}1}=\dfrac9{47}$

That is, we want any 1 spade from the 9 spades available; we don't want any of the other 38 non-spades; and we're drawing 1 card from a total of 47.