Dois eventos dependentes

Dependent events

In contrast to independent events, when events are dependent, the outcome of one has an effect on the outcome of the other.

Sometimes, the probabilities of dependent events are called conditional probabilities. To solve these problems, you'll be calculating the probability of one event given that the other has already happened.

The fancy math way to write this is:

Here, $P(B|A)$ represents the probability of event $B$ given event $A.$ Let's look at an example.

If we don't put the marbles back into the bag, there's one less marble left when we draw the next one. This means that the probability of the second drawing is affected by (i.e., dependent on) the first one.

Let's say I want to determine the following conditional probability.

How did we get than answer? Check out the equation below. It's the standard equation for calculating the probability of one event.

(Video) Probability of Dependent Events

by MathHelp.com

MathHelp.com has a great video on how to find the probability of two dependent events.

A common type of dependent probability is choosing without replacement. The key for doing these problems is making sure you take away from the right value in the formula for the probability of the event. The denominator is always the total number of things you are choosing from, so if you take one thing away and do not replace it, then the next thing you choose will be coming from a smaller total, so the denominator is smaller.

If you are choosing with replacement, on the other hand, then the denominator does not change.

Dependent Events and Conditional Probability

Recall that for two independent events $A$ and $B,$ the probability of event $A$ does not affect the probability of event $B$ and vice versa.

For two dependent events, the probability of one event is affect by the other event. Two events are dependent if the outcome or occurrence of the first event affects the outcome or occurrence of the second event, so that the probability is changed.

**What are some examples? **

In this case, whether event $B$ happens is dependent on event $A.$ If it is sunny and warm today, I will go on a bike ride. A lot of events in life and like this, whether one thing happens relies on the occurrence of another event.

Here is another example.

The chance of Dan getting a chocolate croissant is then dependent on the kind of croissant Sal takes and eats, since we have a specific amount of chocolate and raspberry croissant.

Let's take a look at the notation of this kind of situation.

Let's say we have two dependent events $A$ and $B.$

If the probability of event A is dependent on event $B,$ when we talk about the probability of $A,$ we say the probability of event $A$ given event $B$ is $P(A|B)$.

If the probability of event $B$ is dependent on event $A,$ when we talk about the probability of $B,$ we say the probability of event $B$ given event $A$ is $P(B|A)$. In general $P(A|B)$ and $P(B|A)$ are not the same.

When the probability of one thing is dependent on another, we call it conditional probability. Go here to learn more about conditional probability and how to calculate it.

Try the questions below!

Tossing a $6$ the second time is not dependent on the occurrence of tossing a $6$ the first time. Therefore, these two not not dependent events.