Find the probability of getting the following hands in a game of traditional 5 card poker.
\(\textbf{1)}\) P(Having exactly 2 Aces) (other 3 cards can be anything but aces) The probability is \(\displaystyle\frac{103,776}{2,598,960} \approx 3.9930\%\)
\(\,\,\,\,\,\,\displaystyle \frac{{}_{4}C_{2} \cdot {}_{48}C_{3}}{{}_{52}C_{5}}\)
\(\,\,\,\,\,\,\displaystyle \frac{6 \cdot 17{,}296}{2{,}598{,}960}\)
\(\,\,\,\,\,\,\)The probability is \(\displaystyle\frac{103{,}776}{2{,}598{,}960} \approx 3.9930\%\)
\(\,\,\,\,\,\,\displaystyle \frac{{}_{4}C_{2} \cdot {}_{48}C_{3}}{{}_{52}C_{5}}\)
\(\,\,\,\,\,\,\displaystyle \frac{6 \cdot 17{,}296}{2{,}598{,}960}\)
\(\,\,\,\,\,\,\)The probability is \(\displaystyle\frac{103{,}776}{2{,}598{,}960} \approx 3.9930\%\)
\(\textbf{2)}\) P(Royal Flush) (aka 10, J, Q, K, A and all the same suit) The probability is \(\displaystyle\frac{4}{2,598,960} \approx 0.000154\%\)
\(\textbf{3)}\) P(Straight Flush) (aka same suit, in consecutive number order) The probability is \(\displaystyle\frac{36}{2,598,960} 0.00139\approx 0.00139\%\)
\(\textbf{4)}\) P(Flush) (aka all cards in the same suit.) The probability is \(\displaystyle\frac{624}{2,598,960} \approx 0.0240\%\)
\(\textbf{5)}\) P(Straight) (aka all cards in consecutive number order.) The probability is \(\displaystyle\frac{3,744}{2,598,960} \approx 0.1441\%\)
\(\textbf{6)}\) P(Full house) (aka 3 of a kind and a pair) The probability is \(\displaystyle\frac{5,108}{2,598,960} \approx 0.1965\%\)
\(\textbf{7)}\) P(Four of a kind) The probability is \(\displaystyle\frac{10,200}{2,598,960} \approx 0.3925\%\)
\(\textbf{8)}\) P(Three of a kind) The probability is \(\displaystyle\frac{54,912}{2,598,960} \approx 2.1128\%\)
\(\textbf{9)}\) P(Two pair) The probability is \(\displaystyle\frac{123,552}{2,598,960} \approx 4.7539\%\)
\(\textbf{10)}\) P(One Pair) The probability is \(\displaystyle\frac{1,098,240}{2,598,960} \approx 42.2569\%\)
\(\textbf{11)}\) P(No pair) The probability is \(\displaystyle\frac{1,302,540}{2,598,960} \approx 50.1177\%\)
See Related Pages\(\)
\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Probability with Coin Tosses}\)
\(\,\,\,\,\,\,\,\,\text{Prob(3 heads)}=\frac{1}{8}\)
\(\bullet\text{ Probability with Marbles }\)
\(\,\,\,\,\,\,\,\,\text{Prob(3 red)}=\frac{7}{20}…\)
\(\bullet\text{ Probability with Dice}\)
\(\,\,\,\,\,\,\,\,\text{Prob(Two 6’s)}=\frac{1}{36}…\)
\(\bullet\text{ Probability with Round Tables}\)
\(\,\,\,\,\,\,\,\,(n-1)!…\)
\(\bullet\text{ Probability with Poker Hands}\)
\(\,\,\,\,\,\,\,\,\text{P(Full House)}=…\)
In Summary
Probability is an important concept in the game of poker, as it is used to calculate the likelihood of certain events occurring during the game. For example, probability can be used to determine the chances of being dealt a certain hand, such as a royal flush or a full house. It can also be used to calculate the chances of a certain card being drawn on the flop, turn, or river, or to determine the likelihood of a player winning the hand based on the cards they are holding.
To calculate probabilities in poker, you need to consider the total number of possible outcomes and the number of favorable outcomes for the event you are trying to predict. For example, if you want to calculate the probability of being dealt a royal flush, you would first need to determine the total number of possible hands that can be dealt, which is 52 choose 5 (the number of ways to choose 5 cards from a deck of 52 cards). Then, you would need to determine the number of ways to be dealt a royal flush, which is 4 (there are 4 suits in a deck of cards, and each royal flush must consist of cards from the same suit). The probability of being dealt a royal flush is then 4 / 2,598,960, which is a very low probability.
There are many other probability calculations that can be performed in the game of poker, depending on the specific situation and the information available. By understanding the basic principles of probability, players can use this knowledge to make more informed decisions during the game and improve their betting strategy.