A或B的概率

Finding the Probability of $A$ or $B$

Finding the probability of either event $A$ or event $B$ occurring is simpler than you might think.

When doing problems like these, the 2 events we're interested in should be mutually exclusive. This means that it is not possible for the 2 events to occur at the same time.

This basically means that the probability that either $A$ or $B$ occur is the probability of the union of $A$ and $B$. Think of the $\cup$ as "or."

If you're more of a visuospatial learner, consider the Venn diagram below. When we are trying to find $P(A \cup B)$, we are trying to find the probability of landing in any of circle $A$ or any of circle $B$.

So, if we're trying to find the probability of landing in any of circle $A$ or any of circle B, we can re-write $P(A \cup B)$ as $P(A) + P(B)$.

Let's try an example to see how this works.

Now that we've done most of the legwork, let's come back to the original question:

Often times, instead of being focused on only one possibility, we want to know the probability of several possibilities at the same time. This is where we want to use the "And" and "Or" rules of probability. Specifically in this section, we will focus on the "Or" rules

"Or" is used when you don't care which of the two happens, as long as at least one of them does happen. If, for example, you would win a card game if you drew either an ace or a king, that's a situation in which you would use the "Or" rule. Most of the time, if you would use the word "or" in a description of the problem, you're going to use the "Or" rule.

Usually we denote "Probability of either $A$ or $B$" with the symbol $P(A \cup B)$

The probability for two disjoint events (which cannot happen together) is

$P(A \cup B) = P(A) + P(B)$

This makes some intuitive sense - the probability of either event occurring ($\frac{1}{3}$) is the sum of their probabilities ($\frac{1}{6} + \frac{1}{6}$).

But this doesn't apply to all events, because some events can happen together.

Take a look at this diagram - it graphically represents the probabilities of $A$ and $B$ as area within the sample space:

But what if the two weren't disjoint, and could happen together?

If the shaded region is $P(A \cup B)$, then you can see that by just adding $P(A) +P(B)$, we're double counting the probability of A and B. Thus, the full equation for $P(A \cup B)$ is

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

($P(A \cap B)$ is the "Probability of A and B occurring together")

(Video) Probability and Odds with Shaq

by Tyler Stubenhofer

Tyler Stubenhofer has a great video explaining the relationship between probability and odds.

$\large{\text{Summary}}$

Probablity is $\frac{\text{successes}}{\text{successes + failures}}$, or $\frac{\text{successes}}{\text{total}}$

Odds is represented by the ratio $\text{successes}:\text{failures}$

In these two formulas, the numerators are the same, but the denominators are different.