is any occurrence that has a probability attached to it—in other words, an event is any outcome where you can say how likely it is to occur. But the Law of Large Numbers indicates that over a long period of time, the probabilities and the actual results will start to mirror each other.
Sometimes you can add together probabilities, but it doesn’t work in all circumstances. On the other hand, if you knew P and P but didn’t know how many different colors there were, you’d have to calculate the probability by adding together P and P. We can use the probabilities we already รูเล็ต know to work out the one we don’t know. Probabilities are only indications of how likely events are; they’re not guarantees. But sometimes they can be an useful tool for visualizing what’s going on with probabilities. You’ll see more situations where this helps you later on.
We might not be able to count on being able to do this probability calculation in quite the same way as the previous one. Try the exercise on the next page, and see what happens.
50 sports enthusiasts at the Head First Health Club are asked whether they play baseball, football or basketball. 50 sports enthusiasts at the Head First Health Club are asked whether they play baseball, football, or basketball. If two events intersect, it’s possible they can occur simultaneously. If two events are mutually exclusive, only one of the two can occur.
When you work out P, you can make no such assumption. If you’re given a series of probabilities, put them onto the tree in the relevant position. Try and work out the different levels of probability that you need.
Bayes’ Theorem is one of the most difficult aspects of probability. To find P, we use the same process that we used to find P earlier; we need to add together the probabilities of all the different ways in which the event we want can possibly happen. If we can find what the probabilities of P(Black ∩ Even) and P are, we’ll be able to use these in the formula to calculate P(Black | Even). All we need is some mechanism for finding these probabilities. Let’s start off by looking at the overall probability we need to find, P(Black | Even). When you calculate P(A | B), you have to assume that event B has already happened.
As an example, if you’re given a probability for P(A | B), you’ll probably need the first level to cover B, and the second level A. Probability trees don’t just help you visualize probabilities; they can help you to calculate them, too. Don’t worry, there’s another sort of diagram you can use—a probability tree. Venn diagrams aren’t always the best way of visualizing conditional probability. The probability of A give that we know B has happened. Finding probabilities for multiple intersections can sometimes be tricky. We suggest that if you’re in doubt, draw a Venn diagram and take a good, hard look at which probabilities need to be added together and which need to be subtracted.
In set theory, the possibility space is equivalent to the set of all possible outcomes, and a possible event forms a subset of this. You don’t have to already know any set theory to use Venn diagrams to calculate probability, though, as we’ll cover everything you need to know in this chapter. This event is actually impossible—there is no pocket labeled 38. Very often, the numbers themselves aren’t shown on the Venn diagram. Instead of numbers, you have the option of using the actual probabilities of each event in the diagram. It all depends on what kind of information you need to help you solve the problem. It’s a shorthand way of referring to all of the possible outcomes.