Poker Probability – Part 2: Mutually Exclusive, Independent, and Dependent Events in Poker

In poker, the probabilities we calculate are generally combinations of events. For example calculating the probability that we’ll be dealt aces preflop is actually calculating the combined probabilities of:

• Being dealt an ace of any suit first
• Being dealt a differently-suited ace given that a particular suited ace has been dealt

Each of these is its own probability event, and requires separate calculation before we can know about our aces.

There are three basic types of events that are useful to analyze in poker probability: mutually exclusive, independent, and dependent events. If we want to calculate accurate probabilities in poker, we’ll need to know about all three types.

What Are Mutually Exclusive Events?

Two or more events are mutually exclusive if they can’t occur simultaneously. Consider dealing out starting hands, for example. What are the chances of your first card being any queen in the deck?

This probability breaks down into four possible events. You might be dealt:

• Queen of spades
• Queen of hearts
• Queen of clubs
• Queen of diamonds

You will certainly not be dealt a card that is both the queen of spades and hearts. So these events are mutually exclusive. If one occurs, then by definition no other can.

It’s easy to calculate a probability built on mutually exclusive events. All we need to do is figure out each event’s chance of happening, and then sum the result for all the events.

Using the example above, we notice that there are 52 possible outcomes for our first card dealt. This number just corresponds to the number of cards in a full deck. The chance that we’re dealt one specific card is thus 1/52.

All four possible events are equally likely in our example, so they share the same probability. All we need to do to figure the chance of being dealt a queen is add the probabilities:

P(Dealt a queen) = 1/52 + 1/52 + 1/52 + 1/52 = 4/52 = 1/13

Notice that since all outcomes are equally likely, we could have simplified our equation by multiplying:

P(Dealt a queen) = 4(1/52) = 1/13

What Are Non-Mutually Exclusive Events?

Not all events in poker probability are mutually exclusive. These types of events can occur simultaneously.

For example, maybe we want to calculate the likelihood of drawing a card of a particular rank and suit. In this case, we have two factors:

• Rank
• Suit

Each factor has its own probability, and we’ll need to combine them to find the chance of drawing a particular suited card. Since a drawn card must be both a specific rank and suit, the two events are non-mutually exclusive.

What Are Independent Events?

Events that combine non-mutually exclusive probabilities break down into yet two more event types:

1. Independent events
2. Dependent events

Two or more events are independent if the probability of one occurring does not depend on the probability of the other occurring. In other words independent events do not affect each other’s probabilities.

Calculating the probability of independent events is slightly different from calculating the probability of mutually exclusive events. We can’t just add the probabilities of each event together. Instead, we must multiply the probability of each event. Doing so gives us the joint probability of the two events occurring.

Continuing with our above example, let’s calculate the probability of drawing the four of hearts from a deck of cards. This is actually calculating the joint probability of drawing a 4 and drawing a heart. So we need to know the probabilities of drawing each:

• A 4
• A heart

We know that there are four cards of each rank in a deck of cards. Thus the probability of our card being a 4 is 1/13. We also know that there are thirteen cards of every suit in a deck. Thus the probability of our card being a heart is 13/52, or 1/4.

Now we just multiply these two probabilities to obtain the probability of drawing the four of hearts:

P(4 of hearts) = P(Card is a 4) * P(Card is a heart)

P(4 of hearts) = (1/13) * (1/4)

P(4 of hearts) = 1/52

What Are Dependent Events?

Some poker probabilities do influence each other. Two or more events like this are called dependent events, because the probability of one depends on the probability of the other.

When we want to calculate the probability of two dependent events occurring, we’re looking for the conditional probability. That is, the probability of an event A occurring given that event B occurs as well.

We use a special notation when calculating conditional probabilities. Say we want to know the probability of event B happening if A also happens. We would write the following:

P(A | B)

And we’d express it as “the probability of event B given event A”.

Dependent Events in Poker: Probability of Flopping a Straight

For example, let’s say we want to calculate the probability of a connected hand flopping its high straight or straight flush. Specifically, we have T-J and want to hit Q-K-A for the nuts. We need to calculate three separate probabilities here. Namely, that:

A. The first flop card is either a Q, a K, or an A.
B. The second flop card one of the two cards that did not fall for event A.
C. The third flop card is the remaining card we need to make our straight.

Calculating the probability of event A is simple. There are four Qs, four Ks, and 4 As in the deck, which makes 12 cards that help us. There are a total of 50 unseen cards left in the deck. So:

P(A) = 12/50 = 6/25

Now we need to calculate the probability of the second flop card being one of the two remaining ranks that help us. We labeled this earlier as event B. The probability of event B depends on event A having already happened, so we write it P(B | A).

P(B | A) = 8/49

Now we need to figure out the probability of the third flop card being the final rank that makes our straight. We called this event C earlier. We write this as P(C | (A and B)), since the probability of C happening depends on events A and B having already occurred.

P(C | (A and B)) = 4/48 = 1/12

Now we multiply the probabilities of these dependent events together to obtain the conditional probability of A, B, and C:

P(A and B and C) = (6/25 * 8/49) * 1/12
P(A and B and C) = 0.003265306

We’ve converted the solution to decimal form, since the answer in fraction form deals with very large numbers. Our calculation tells us that the probability of flopping a specific straight or straight flush with connected hole cards is 0.003265306. This translates roughly to 0.33%.

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