## Related questions with answers

More poker hands. a. What is the probability that a poker hand is a three of a kind? (A three of a kind has three cards of the same value and two other cards of different values, such as three 10s, a 7, and a jack.) b. What is the probability that a poker hand is a full house? (A full house has three cards with one common value and two other cards of another common value, such as three queens and two 4s.) c. What is the probability that a poker hand has one pair? (One pair means two cards have the same value and three other cards have three other values, such as two 9s, a king, an 8, and a 5.) d. What is the probability that a poker hand has two pairs? (Two pairs means two cards have one common value, two more cards have another common value, and a fifth card has yet another value, such as two jacks, two 8s, and a 3.)

Solution

VerifiedDEFINITIONS

A $\textbf{standard deck of cards}$ contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs) and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are the jacks J, queens Q and kings K.

Definition $\textbf{binomial coefficient}$ (order is not important when selecting $r$ objects from a set of $n$ objects):

$\left(\begin{matrix}n\\ r\end{matrix}\right)=\dfrac{n!}{r!(n-r)!}$

with $n!=n\cdot (n-1)\cdot ...\cdot 2\cdot 1$.

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