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.

Short answer

The probability of completing the flush in spades is \( \frac{3}{16} \).

Step-by-step solution

Understanding the Card Deck

A standard deck of cards contains 52 cards: 4 suits (hearts, diamonds, clubs, spades) each of 13 cards (Ace to 10, Jack, Queen and King). So, there are 13 spades.

Replacing the Non-Spade Card

In the game, four spades are already in hand, leave 9 spades in the deck. The deck has now 48 cards (52 original cards - 4 spades in hand) from which we will draw a card.

Calculating the Probability

The probability is calculated by dividing the number of successful outcomes by the total outcomes. Here, successful outcome is drawing a spade. So the probability (P) can be calculated as P = Number of Spades Remaining / Total Number of Cards Remaining. That gives \(P = \frac{9}{48} = \frac{3}{16}\).