In this explainer, we will learn how to calculate conditional probability using formulas and Venn diagrams.
In probability, there are situations where new information changes the probability of events. For instance, say that we are trying to guess a card that our friend has selected out of a deck of 52 cards. Based on pure speculation, we can guess that our friend has selected the ace of spades. Since there is no information about which one is the correct card, the probability that our guess is correct is . What if our friend reveals that the correct card is a spade? Given this new information, the probability that our guess is correct will increase to .
We can note that the value of the probability that our guess is correct changed when new information was revealed. This demonstrates the concept of conditional probability, which is the probability of an event given a specific outcome of another event. In this context, the probability represents the conditional probability that the correct card is the ace of spades given that the correct card is a spade.
Definition: Conditional Probability
Let and be events in the same sample space. The conditional probability of given , denoted , is the probability of event if we already know that event is true.
To compute the conditional probability, we can either use a formula or a Venn diagram. Let us begin with the approach using a Venn diagram, which can be used to derive our formula. Consider the Venn diagram below representing events and in the sample space .
The number inside each region of the Venn diagram represents the number of possible outcomes of the corresponding event. Let us further assume that each outcome is equally likely.
Using this Venn diagram, we will find the conditional probability . If we already know that event is true, event will be true only for the outcomes in the intersecting region, which is the pink region in the Venn diagram. This means that, with this given information, there are 2 different outcomes for which event is true. Also, since we know that event is true, there is a total of different possible outcomes. This leads to the conditional probability
In our first example, we will demonstrate this process for a real-world problem.
Example 1: Conditional Probability
On the street, 10 houses have a cat (C), 8 houses have a dog (D), 3 houses have both, and 7 houses have neither.
- Find the total number of houses on the street. Hence, find the probability that a house chosen at random has both a cat and a dog. Give your answer to three decimal places.
- Find the probability that a house on the street has either a cat or a dog or both. Give your answer to three decimal places.
- If a house on the street has a cat, find the probability that there is also a dog living there.
Answer
Before considering the questions, let us organize the given information into a Venn diagram. Since we are told that 3 houses have both cats and dogs, we can fill in the intersecting area of our Venn diagram first.
Now, if 10 houses have cats, and we know that 3 of them also have dogs, this leaves houses that have only cats.
Likewise, if 8 houses have dogs, but 3 of those houses also have cats, then houses just have dogs.
Finally, we are told that 7 houses have neither cats nor dogs, so we can fill in the area outside the ovals.
We can find the total number of houses by adding the numbers from each section of the diagram. So the total is . Now let us consider each question.
Part 1
This question asks for the probability that a randomly selected house has both a cat and a dog. From the Venn diagram above, we can see that there are 3 houses with both a cat and a dog and 22 houses in total. Since a house is chosen at random, we can assume that each outcome is equally likely. Hence, the probability that a randomly selected house has both a cat and a dog is
Therefore, the probability is 0.136 to three decimal places.
Part 2
This question asks for the probability that a randomly selected house has either a cat or a dog or both. From the Venn diagram, we can see that there are houses that meet this criterion. Since there are 22 total houses, this probability is
Hence, the probability that a house has either a cat or a dog or both is 0.682 to three decimal places.
Part 3
This question asks for the probability that a dog lives in a house if the house has a cat. The phrase “if the house has a cat” restricts the number of possible houses to only the 10 houses that own a cat.
Out of the 10 houses we are now considering, only 3 have dogs, so if a house on the street has a cat, then the probability it also has a dog is .
In the previous example, we computed various probabilities using a Venn diagram. The probability we computed for the final part of the example is actually the conditional probability. If is the event that a house has a dog, and is the event that a house has a cat, then the probability that a house has a dog given that it has a cat is the conditional probability .
By using a Venn diagram to compute the conditional probability, we can see that the given condition restricts the sample space to a subset. “Given that the house has a cat” implies that we only need to consider a subset of 10 houses that have cats. Since we are restricted to houses with cats, a house with a dog in this case would also mean that it has both a cat and a dog and hence belongs to the intersection .
Since each outcome is equally likely, we can represent this conditional probability as
While this produces a formula for conditional probability, this version of the formula is not very useful since we need to count the number of outcomes. If we divide the top and the bottom of this fraction by the total number of houses in the street, we can obtain the probabilities of these events instead. In other words,
Although this formula was obtained using a specific example, it is in fact a general formula that can be used to compute any conditional probability.
Formula: Conditional Probability
Let and be events in the same sample space, where . The conditional probability of given is given by
Let us consider an example where we will compute a conditional probability using this formula.
Example 2: Determining Conditional Probability
Suppose and are two events. Given that and , find .
Answer
Recall that is the conditional probability of given , which can be computed using the formula
Since we are given that and , we can substitute these values into the conditional probability to obtain
Hence, the probability is .
In the next example, we will apply this definition to compute the conditional probability from a Venn diagram containing probabilities of events.
Example 3: Using Probabilities in a Venn Diagram to Calculate Conditional Probabilities
Consider the following Venn diagram.
Calculate the value of .
Answer
Recall that is the conditional probability of given , which can be computed using the formula
We need to identify the probabilities and from the Venn diagram. Since the given Venn diagram contains the probabilities of events corresponding to the regions, we need to simply identify these regions. The probability of the intersection is given in the region that is the overlap of the two circles. This gives .
On the other hand, is the region given by the first circle, which is split into two pieces. We can compute the probability of by adding these two probabilities:
Substituting these values into the conditional probability, we have
In a conditional probability , interchanging the locations of events and to write produces a different conditional probability. In most cases, these two values are not the same, as they represent different probabilities. As we can compute , we can also compute
We need to remember that the denominator of the conditional probability is always the probability of the condition.
In the next example, we will compute both and using this formula.
Example 4: Calculating Conditional Probabilities
It has been found for two events and that , , and .
- Work out .
- Work out .
- Work out .
Answer
Part 1
Let us find the probability of the intersection . We are given and . Recall that represents the complement of , and the complement rule states that
Substituting the given probability , we obtain
Now, we are also given . Recall the general addition rule, which states that
Substituting all the known probabilities,
Rearranging this equation so that is the subject, we obtain
Part 2
Recall that is the conditional probability of given , which can be computed using the formula
Since we know and , we obtain
Part 3
Similarly, the conditional probability can be computed using the formula
Since we know and , we obtain
In the previous example, we computed the conditional probabilities and . We can observe that these probabilities have different values in this example as expected.
We now turn our attention to yet another use of conditional probability. Using the formula , we can obtain another useful formula when we multiply by on both sides of this equation.
Rule: General Multiplication Rule
Let and be events in the same sample space. Then,
This formula tells us that the probability of the intersection of two events is equal to the product of the conditional probability and the probability of the conditioned event.
Let us consider an example where we apply this rule to find the probability of an intersection.
Example 5: Determining the Intersection of Two Events Using the Multiplication Rule
Suppose that and are two events. Given that and , find .
Answer
We recall the general multiplication rule, which states that for any two events and ,
Since we are given that and , we can substitute these values into the general multiplication rule to obtain
In the final example, we will apply the general multiplication rule to a real-world problem to compute a probability.
Example 6: Calculating Conditional Probabilities
Mona either takes the bus to school or, if she misses it, she walks. The probability that she catches the bus on any given day is 0.4. If she catches the bus, the probability that she will get to school on time is 0.8, but if she misses the bus and has to walk, the probability of her being on time drops to 0.6.
- Work out the probability that she catches the bus and is on time for school on a given day.
- Work out the probability that she is on time for school on a given day, whether or not she catches the bus.
- Hence, find the probability that she will be late for school on a given day.
Answer
Before we begin answering the questions, let us first interpret the given information and put it into probability notation. In this example, Mona either takes the bus to school or misses it. Let denote the event that she takes the bus, which means that the event that she misses the bus is denoted by the complement . Also, we can let denote the event that she gets to school on time, which means that the event that she is late to school is given by the complement .
We are given that the probability that she catches the bus on any given day is 0.4, which means
If she catches the bus, the probability that she will get to school on time is 0.8. Since this statement begins with the phrase “if she catches the bus,” this is a conditional probability of her getting to school on time given that she catches the bus. Hence,
If she misses the bus and has to walk, the probability of her being on time drops to 0.6. This is also a conditional probability of event given the complement . Thus,
Now that we have interpreted all the given information and put it in probability notation, let us begin examining each question.
Part 1
We need to find the probability that she catches the bus and is on time for school on a given day. Using the probability notation, this is the probability . We recall the general multiplication rule which states that
Since we know that and , we have
Hence, the probability that she catches the bus and is on time for school on a given day is 0.32.
Part 2
We need to find the probability that she is on time for school whether or not she catches the bus. To compute this probability, let us consider the Venn diagram with circles representing events and .
We have filled in the probability from the previous part, , in the Venn diagram. Also, Since , we know that the probabilities in the region representing event must sum to 0.4. This leads to the probability 0.08 in the diagram above.
Now, we want to compute . For this, we need to first compute the probability for the empty region in the circle representing event . This region represents the event . Applying the general multiplication rule, we can write
We are given that . Also, we can apply the complement rule, , to compute
Substituting these values into the general multiplication rule, we obtain
We can now put this in the Venn diagram.
Summing the probabilities in the region representing event , we obtain
Hence, the probability that she is on time for school whether or not she catches the bus is 0.68.
Part 3
We need to find the probability that she will be late for school on a given day, which is . We can apply the complement rule together with from part 2 to obtain
Hence, the probability that she will be late for school on a given day is 0.32.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- Let and be events in the same sample space. The conditional probability of given , denoted , is the probability of event if we already know that event is true.
- Let and be events in the same sample space, where . The conditional probability of given , is given by
- Let and be events in the same sample space. Then,
