When I was younger my grandma used to spend hours with me trying to teach me to play poker, but really she wasn’t just teaching me to bet, check, call, and all that good stuff; she was teaching me how to bluff. My grandma often participated in poker tournaments when she was younger and somehow she always knew all the possibilities of what my cards could be and if I was bluffing. I thought it was just because she was so good at it but now that I’m older I understand more that my grandma wasn’t just an all-knowing poker player, she’s just really good at math and probabilities (and telling whether or not a kid is lying).

The question I plan on answering in this investigation is whether or not poker is a complicated, mathematical game, based on bluffing and probability or if it is just a card game based purely on luck.Important Poker TermsAnte – the first bet placed by all players (all players place the same amount) in order to be dealt into the gameFold – to get out of a hand. You just throw your cards away, and that’s it. You can do this at any point in the gameCheck – Basically it’s making the next person in the game bet before you, then seeing if you want to call, fold, or raiseCall – Putting in the same bet as another person and moving onto the next roundRaise – Raising someone else’s bet if you believe you have a better hand, and want to win a bigger amountCapped betting – This means you can’t bet anymore, you can only call. This happens after about 4 raises, or when the limit is met Bluff – This is when a player doesn’t have a very good hand, but bets like crazy so it seems like he does and try to get others to foldOut- Any unseen card that, if drawn, will improve a player’s hand to one that is likely to winFlop- If you have two pairs and hope to make a full house on the river, but your opponent already has four of a kind, you are “drawing dead”River- After the flop betting round ends, a single community card is dealt, followed by a third betting round. A final single community card is then dealt, followed by a fourth betting round In order to answer my research question I plan on analyzing poker professionals’ methods and looking at charts of possible outcomes in a poker game and probabilities of winning a hand with a standard 52 card deck.

Information / MeasurementPossible outcomes of a poker game with two players and three cards (Queen, King, and Ace) Flop to turn Turn to River Turn and River 0uts%odds%odds%odds12.146-12.245-14.322.26-124.

322.5-14.322-18.410.9-136.814.67-16.

514.33-112.57.00-148.510.

75-18.710.5-116.

55.06-1510.68.

4-110.98.2-120.

33.93-1612.86.83-1136.

67-124.13.15-1714.95.71-115.25.57-127.

82.6-18174.88-117.44.75-131.52.17-1919.14.

22-119.64.11-1351.86-11021.33.

7-121.73.6-138.41.6-11123.43.

27-123.93.18-141.71.4-11225.

52.92-126.12.

83-1451.22-11327.72.62-128.32.54-148.11.08-11429.

82.36-130.42.29-151.2.

95-11531.92.13-132.62.07-154.1.

85-116341.94-134.81.88-157.75-11736.21.

77-1371.71-159.8.67-11838.31.61-139.11.56-162.

4.6-11940.41.47-141.31.42-165.

54-12042.61.35-143.51.

3-167.5.48-1Mathematical ProcessThe first diagram shows the outcomes of a poker game with three cards (Queen, King, Ace) and two players, each player is dealt one card and can place bets or fold from there. The X represents the next possible move in the game, bet or fold and whether or not to fold from there.

The b stands for bets/bluffing and the c for call, I used the player that was dealt a King for an example, but there can be more bets made until a player calls. The probability of the card the Players are dealt can be modeled by this equation? x ½ x b = b/6 If the bet is simply one dollar the expected value for the players would bePlayer 1 = 0.5b(c-1/3) +/- c/6Player 2 = -0.5b(c-1/3) +/- c/6 You can also calculate expected values of how much you will earn through the diagram, depending on the bets made. E(P) represents the expected earnings of a playerE(P)= 1/6bc – 1/3b(1-c) – 1/6c 1/6b= 1/2b (c – 1/3) -1/6c The amount that one of the players will win is just the negative value of the opposing player, more simply put, if Player 1 is E(P) and Player 2 is E(A) then E(P) is -E(A) because Player 2’s loss is Player 1’s gain, the player would want to minimize their opponent’s expected earnings as much as possible, you can do so using this equationE(P)= 1/2c (1/3 – b) + 1/6bThe expected value of the opponent would be at its minimum when b=0. This can be achieved by never bluffing but it would take longer. The only way for a player to defend himself against being exploited this was is by using this expressionE(P) = 1/2c (1/3 – b) + 1/6b If one or both player use the strategy of never bluffingE(P) = 1/18