The probability of an event is how often it will happen compared to how often it won't.
This is the short version. It's not formally correct and it's not the whole truth, but it's a good start.
Percentage or decimals
We often express probabilities as a percentage. Flipping a coin and getting tails is a 50% probability. We'll get tails half of the time, and fifty percent is half of 100 percent.
Another way of expressing probabilities is as a decimal number between 0 and 1. Expressed like this, flipping a coin and getting tails is a 0.5 probability, since 0.5 is halfway between 0 and 1.
This way of expressing probabilities is the one used in probability calculations. Both ways of writing are perfectly okay, and switching between them should be easy with a little practice.
Example: rolling a die
When you roll a die, what is the probability of rolling a six?
Well, the die is symmetrical, so all sides are equally likely to come up. In other words, they all have the same probability. Let's call this probability p (a number between 0 and 1).
At the same time, we know that one of the sides must come up, so the sum of their probabilities is 1.
These two things put together tell us that p+p+p+p+p+p=1. We conclude that p = 1/6. Expressed as a percentage, this is close to 17%.
This is the probability for each of the sides, so it's true in particular for the side with six eyes.
Thus, the probability of rolling a six is 1/6, or 17%.
Example: picking a card
If we deal one card from a shuffled deck, all cards are equally likely to come up. Since there are 52 cards in the deck, the probability of dealing a particular card is 1/52, which is pretty close to 2%.
Example: Picking a spade
If we pick one card from a full deck of cards, how likely is it to be a spade?
Solution: There are 52 cards in the deck, 13 of them are spades. Each of them comes up with probability 1/52, as we saw above. If you add up all these probabilities you arrive at 13/52 = 1/4.
Thus, the probability of picking a spade is 1/4, or 0.25, or 25%.
Combined probability for two events
Often you need to figure out the probability for two events to occur - both of them. If you know the probability of each event individually, how can you find their combined probability?
Without going into the technical details, this is how you do it. If one event has probability 0.3 and the other 0.5. The probability that both will occur is 0.3 * 0.5 = 0.15.
To get the answer, you just multiply the probabilities.
Combined probability for either or
Another common question is the probability that either of two events occur.
For example, if you have both a flush draw and a straight draw, what is the probability that you hit one of them?
In this case it's a little bit more complicated. To get the combined probability for the two events, you have to use this formula:
P = 1 - (1 - x)*(1 - y)
Example: If one event has probability 0.3 and the other 0.5, the probability that at least one of them occur is 1 - (1 - 0.3)(1 - 0.5) = 0.65.
The probability that at least one of them will occur is 0.65, or 65%.
Note that when it's either or, the combined probability is greater than for any of the involved events as such.
On the contrary, as we saw above, the probability that both events occur is always lower than the individual probability for any of the two events.
/Charlie River
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This article is part of the poker math series:
- Basic Probability Theory
- Where do Probabilities Live?
- Average, Expected Value, Variance and More
- What Is Odds?
- What is Outs in Poker?
- The Math behind Calling and Folding
- EV - How To Calculate It
- Variance - How to Calculate It
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