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4. Understanding Conditional Probability
Conditional probabilities are useful when the probability of one event depends on the outcome of another event. Learn how to use conditional probabilities in this lesson.
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Lesson Goal (00:26)
The goal of this lesson is to learn how to calculate conditional probabilities.
Dependent and Independent Events (01:49)
Dependent events are events where the probability of one event is affected by the outcome of another event. For example, an ad campaign for a product is more likely to increase sales for the product if the price of the product is discounted. This is the opposite of independent events, where the outcome of one event does not affect the probability of the other event.
Understanding Conditional Probabilities (02:05)
A conditional probability is the probability of one event occurring given that another event has occurred. We denote this probability using a vertical bar, for example P(A|B) is the probability of event A occurring given that event B has already occurred. Note that the conditional probability of A given B (A|B) may not be the same as the conditional probability of B given A (B|A).
If events A and B are dependent, then the conditional probability will be different from the individual event’s probability. For example, if the probability of A given B is different from the probability of A, this tells us that whether event B occurs influences the probability of event A, which means the events are dependent.
Probabilities of AND Events (03:20)
For two independent events, we can find the probability of both occurring by multiplying their individual probabilities. If the events are dependent, then the probability for the second event should be replaced by the conditional probability given that the first event has occurred. These conditional formulas can also be used for independent events.
This indicates how we can use conditional probabilities to identify if events are dependent or independent. For two independent events, the probability of each event is equal to the conditional probability given that the other event has occurred. If this is not the case, the events are dependent.
In the previous lesson, we learned how to calculate the probabilities of multiple events occurring.
We learned that the probability of one event and another event occurring is affected by whether or not the events are independent.
To calculate the probabilities for multiple dependent events we need to use conditional probability.
In this lesson, we'll learn how to calculate conditional probabilities.
We'll consider the problem of a supermarket selling many products. They have two main methods of increasing sales of a product advertising the product or discounting the product.
The supermarket wants to understand the probability that advertising a product will increase its sales.
They analyze past ad campaigns and find 60% of them led to an increase in sales of the product.
This suggests the probability that advertising increases sales of a product is 0.6.
However, we suspect that our product is more likely to see an increase in sales if it's also discounted at the time of the ad campaign. Let's define this problem in terms of events.
Event A is the event that an ad campaign for a product increases its sales.
Event B, is the event that the product is discounted at the time of the ad campaign.
We know the probability of A is 0.6, but we believe that this probability is affected by the outcome of Event B.
As a result, we can describe A and B as dependent events.
Two events are dependent if the probability of one event is affected by the outcome of another event, this is the opposite of the independent events that we saw in the previous lesson. When we have dependent events, such as this, we need to consider conditional probabilities.
A conditional probability is the probability of one event occurring given that another event has already occurred.
For example, let's assume that we want to know the probability that an ad campaign increases sales for a product if that product is already discounted. In terms of our events, we say, this is the probability of Event A occurring given that Event B has occurred. We write this as the probability of A given B with the vertical bar indicating a conditional probability. Our supermarket analyzes past ad campaigns and finds the 80% of ad campaigns where the product was discounted led to an increase in sales for that product.
As a result we can say that the probability of A given B is 0.8. In this case, the probability of A given B is different from the probability of A.
This indicates that Event B does affect event A which confirms that the events are dependent.
Now that we understand what conditional probability is, let's learn how it can affect the probability of two events of interest, both occurring.
We previously saw that the probability of two events occurring could be found by multiplying the probabilities of the individual events. However, this only applies to independent events.
For dependent events, the probability of events A and B occurring is a probability of A times the conditional probability of B, given A.
Alternatively, we can reverse the order of the events and say that the probability of A and B is the probability of B times the conditional probability of A given B.
In our case, if we wanted to find the probability of an ad campaign boosting sales for a product and the product being discounted, we would multiply the probability of the product being discounted by the product of an ad campaign boosting sales, given the product is discounted.
We can also use these formulas for independent events.
As we can see if A and B are independent, then the probability of A and B can be calculated in three different ways.
By comparing these three formulas mathematically, we can identify the conditions that tell us when events are dependent or independent.
For independent events, we would find that the probability of B is the same as the probability of B given A.
Similarly, the probability of A is equal to the probability of A given B.
This simply tells us that the probability of A is not effected by the outcome of B. And the probability of B is not affected by the outcome of A.
For dependent events we can say that the probability of A is not equal to the probability of A given B. And the probability of B is not equal to the probability of B given A. This indicates that each event does not affect the other meaning they are independent.
Let's stop the lesson here.
In the next lesson, we'll look at a common application of conditional probability known as Bayes' Theorem.