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Edutainment – Pick the Winner 2 of 4

Nov 22, 2007 Listen up edutainers!
Today we’re going to talk about Probabilities and Odds. When discussing probability it is useful to start by defining a set terminology, in order to avoid confusion stemming from fuzzy everyday usage of mathematical terms.

Hence, let’s start by defining the term “experiment”: an experiment is any action which has a number of randomly determined outcomes, such as flipping a coin, rolling a die, or drawing a card from a deck. Let us also define the term “event” to be any set of outcomes from an experiment, satisfying some predefined condition; for instance, an event could be “rolling a six”, “flipping heads”, etc.

Events can also be compound, such as “rolling an even number” or “drawing a heart”. The probability of an event is equal to the number of outcomes which fulfill the predefined condition, divided by the total number of outcomes of the experiment. This last point is very important in poker, and often misunderstood. Consider a common situation: you have two suited cards in the hole, and there are two more cards of the same suit on the board. In other words, you have a flush draw. What is the probability of drawing another card of that suit? Instinctively you might say 25 %, since there are 4 different suits and one of these is hearts. Hence, 1 of 4 which is 25 %.

This is wrong! In this scenario, the experiment is “drawing another card”, and the event we want to evaluate is “drawing another heart.” Because you have seen five cards (the two in your hand and the three on the board), there are 47 unseen cards, which constitute all the possible outcomes of the experiment. If you have two hearts, and there are three cards on the board of which two are hearts, that means only nine more hearts remain in the deck (13 minus 4), constituting the outcomes which fulfill the events condition. Hence, the probability of the event “drawing another heart” is 9/47, or approximately 19 %.

Another issue that intuitively might seem very confusing is the difference between “cards remaining in the deck” and “unseen cards”. Consider if you are playing a round of Texas Hold’em against three friends. You are each dealt two cards to start, and then another three community cards are dealt on the flop. This means that 2 x 4 + 3 = 11 cards have been dealt from the deck. Hence, only 41 cards physically remain in the deck. So now what is the probability of drawing a fifth heart on the turn?

It is still 9/47, since your opponents cards are still unknown to you, and must therefore be considered as possible outcomes of the experiment. “Unseen cards” and “cards remaining in the deck” are not the same thing, and when estimating probabilities, we use the former.

That was all for today. But please just hold on. Tomorrow we’ll continue talking about Odds vs. Probabilites. In the meanwhile, why not try out the mini game perfectly suited for this subject – Pick the Winner.

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