Get a better hand

Probability Level 4

You are playing a game of poker , and you are dealt the following hand of cards from a shuffled standard poker deck:

$\text{A}\spadesuit, \text{A}\clubsuit, \text{A}\color{#D61F06}{\heartsuit}, 6\color{#D61F06}{\heartsuit}, 10{\color{#D61F06}{\diamondsuit}}.$

You put the $6\color{#D61F06}{\heartsuit}$ and $10\color{#D61F06}{\diamondsuit}$ cards aside, and request to be dealt two new cards.

What is the probability that you will improve your hand to a Four of a Kind or a Full House ?

Round your answer to three decimal places.

Note : Cards dealt to you are no longer in the deck. The $6\color{#D61F06}{\heartsuit}$ and $10\color{#D61F06}{\diamondsuit}$ cards are put aside; they are not put back into the deck.

The answer is 0.104.

1 solution

Andy Hayes
Jul 5, 2016

Relevant wiki: Math of Poker - Basics

There are 47 cards left in the deck. There are $\binom{47}{2}=1081$ possible combinations of 2 cards to be dealt to you.

The number of ways to improve your hand to a Full House is:

$\binom{10}{1}\binom{4}{2}+\binom{2}{1}\binom{3}{2}=66$

Explanation: There are 10 ranks that are not A , 6 , or 10 . Select one of those ranks, then select 2 suits for those cards. Alternatively, choose 1 rank from either 6 or 10 , then choose 2 suits for those cards from the remaining 3 suits.

The number of ways to improve your hand to a Four of a Kind is:

$\binom{1}{1}\binom{46}{1}=46$

Explanation: There is only 1 Ace card left, and you must select it. Then, you can select any of the 46 remaining cards.

These combinations are disjoint, therefore they can be added together to obtain the total number of ways to improve your hand.

$66+46=112$

Thus, the probability to improve your hand is $\frac{112}{1081}\approx \boxed{0.104}$ .

Get a better hand

The answer is 0.104.

1 solution

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