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Applications of Probability

The Relationship Between Independence and Conditional Probability

In the previous lesson, the independence of events was examined. In this lesson, the conditional probability and its connection with independence will be discovered through the use of real-world examples.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Analyzing Connections Between Events

Vincenzo is thinking about taking a trip overseas. He is having a tough time deciding between France and Antarctica. In France he can visit the Eiffel Tower — he studies structural engineering — but Antartica has penguins and Vincenzo really loves penguins!
1. Tearrik next to the Eiffel Tower. 2. Tearrik in Antartica. 3. Tearrik and a penguin
Vincenzo pulls up a search engine and types "France vs. Antartica, I wanna see Penguins." He finds an interesting survey that just so happens to contrast France and Antarctica. The survey even asks specifically about penguins!
  • Of people surveyed, people prefer going to France over Antarctica.
  • Of those only five people saw penguins.
  • Of the people who traveled to Antarctica, only three of them did not see penguins.

The remaining results from the survey are organized in the following table.

Frequency table comparing people who go to France/Antarctica and see a penguin with those who do not

Consider the presented data to find the probabilities of the following scenarios.

a Vincenzo decides to travel to France. What is the probability that he will see a Penguin? Round the probability to the nearest percent.
b Yesterday, Vincenzo saw a penguin! What is the probability that he was in France? Round the probability to the nearest percent.
c Vincenzo’s trip just ended. Sadly, he did not see a single Penguin. What is the probability that he chose to go to Antarctica? Round the probability to the nearest percent.

Example

Relationship Between Dependent Events

Diego wants to pick two books at random from a pile of five books. Three of them are Geometry books, and the other two are History books. Below, the sample space of this situation is shown, where represent the Geometry books, and and represent the History books.
Three Geometry books and Two History books
Imagine standing next to Diego and calculating the probabilities in real time with him. Let be the event that the first book chosen is a History book, and be the event that the second book chosen is a Geometry book. Now, consider the following scenarios.
a Find and Express the probabilities as fractions in their simplest form.
b Find and Express the probabilities as fractions in their simplest form.
c If the first book Diego picked is a History book, what is the probability that the second book will be a Geometry book? Express the probability as a fraction in its simplest form.
d Diego chose both books before anyone else saw. If the second book Diego picked is a Geometry book, what is the probability that the first book he chose is a History book? Express the probability as a fraction in its simplest form.
e Is there any relationship between the probabilities found in Parts B, C, and D?

Answer

a Probabilities:
b and
c
d
e Sample answer:
  • The probability that event happens given that event happened equals
  • The probability that event happens, given that event happened equals

Hint

a Every outcome that has a History book in the first position satisfies Similarly, every outcome that has a Geometry book in the second position satisfies Divide the number of favorable outcomes by the total number of outcomes.
b Divide the numbers found in Part A.
c The sample space of the given situation is not the original sample space. Start by finding the new sample space. Later, determine how many of the outcomes have a Geometry book in the second position.
d The sample space of the given situation is not the original sample space. Also, it is different from the one found in Part C. In this new sample space, count the outcomes that have a History book in the first position.
e Compare the probabilities found in Parts B, C, and D. Write a sentence stating the relationship found.

Solution

a By definition, the probability of an event is found by dividing the number of favorable outcomes by the total number of outcomes in the sample space.

From the given diagram, the sample space contains different outcomes. The required probabilities can be found one at a time.

  • To find divide the number of outcomes in the sample space with a History book in the first position by
  • To find divide the number of outcomes in the sample space with a Geometry book in the second position by
  • To find divide the number of outcomes in the sample space with a History book in the first position and a Geometry book in the second position by
The following diagram summarizes all the computations.
Three Geometry books and Two History books
Notice that That relationship implies that the given events are dependent.
b In this part, the probabilities found in Part A will be used.
Start by finding the ratio of to
Substitute values and simplify
Next, find the ratio of to
Substitute values and simplify
c For this situation, it is known that the first book chosen is a History book. Knowing this fact reduces the number of possible outcomes because some of the outcomes are ignored in the new situation. Therefore, the new sample space contains fewer outcomes compared to the original one.
Three Geometry books and Two History books

As seen, there are outcomes in the new sample space and of them have a Geometry book in the second position. Therefore, the probability that the second book is a Geometry book, given that the first book is a History book, can be found using the following process. Consequently, if the first book chosen by Diego is a History book, the probability that the second book is a Geometry book is

d In this case, it is known that the second book chosen is a Geometry book. This reduces the number of possible outcomes from the original sample space. Therefore, the new sample space contains fewer outcomes and it is different from the one used in Part C.
Three Geometry books and Two History books

As seen, there are outcomes in the new sample space and of them have a History book in the first position. Therefore, the probability that the first book is a History book, given that the second book is a Geometry book can be found as follows. Consequently, if the second book chosen by Diego is a Geometry book, the probability that the first book is a History book is

e Start by writing the four probabilities found in Parts B, C, and D.

Comparing the four probabilities, it can be seen that the first probability found in part B equals the probability found in part C.

The probability that the second book is a Geometry book given that the first book chosen is a History book equals

The previous statement can also be rewritten in terms of and as follows.

The probability that event happens given that event happened equals

Similarly, the second probability found in part B equals the probability found in part D. This leads to write the following relation.

The probability that the first book is a History book, given that the second book is a Geometry book equals

As before, the previous statement can be rewritten in terms of and

The probability that event happens, given that event happened equals

Example

Finding the Probability of Triggering an Alarm

At Troupial Airport, engineers are testing a prototype of a prohibited substance detector. If there is a forbidden item in a bag, an alarm is supposed to be triggered. To test the detector, bags will be checked and of them will randomly contain forbidden items.
Forbidden items detector
If a bag contains a forbidden item, there is a chance that the alarm is triggered. On the other hand, if a bag does not have a forbidden item, there is a chance that it triggers the alarm. Calculate the probabilities of the events described below. Write the probabilities as a percent rounded to two decimal places, when necessary.
a Find the probability that a bag triggers the alarm.
b Find the probability that a bag triggers the alarm and contains a forbidden item. Also, find the probability that a bag does not trigger the alarm and contains a forbidden item.
c If Mark's bag triggered the alarm, what is the probability that his bag contains a forbidden item?
d If Izabella's bag did not trigger the alarm, what is the probability that her bag contains a forbidden item?
e Based on Parts C and D, what can be said about Mark and Izabella situations?

Answer

a
b and
c The probability that Mark's bag contains a forbidden item given that it triggered the alarm is about
d The probability that Izabella's bag contains a forbidden item given that it did not trigger the alarm is about
e Example Solution:
  • There is a chance that Mark's bag contains a forbidden item. However, this probability is not big enough to ensure Mark has a forbidden item. Therefore, it is doubtful — but possible — that Mark contains a forbidden item in his bag.
  • There is a chance that Izabella's bag contains a forbidden item. Since this probability is very small — less than — it is almost certain that Izabella does not have forbidden items in her bag — but still possible.

Hint

a Make a tree diagram and write the number of bags corresponding to every node. The probability of triggering the alarm is the total number of bags that triggered the alarm divided by the total amount of bags checked.
b Determine how many bags contain a forbidden item and trigger the alarm. Divide this number by the total number of bags checked. Similar reasoning can be used for the case where the bags do not trigger the alarm.
c Considering the diagram drawn in Part A, what is the total number of bags that trigger the alarm? Of all these bags, how many actually had forbidden items?
d Of all the bags that did not trigger the alarm, how many had forbidden items? Divide this last number by the total number of bags that did not trigger the alarm.
e If the probability of an event is closer to than then it is doubtful that the event actually happened. If the probability of an event is close to it is almost certain that the event did not happen.

Solution

a To find the probability that a bag triggers the alarm, a tree diagram will be drawn. To do so, first notice that the situation can be divided into three different stages.
Defining three stages
All of the previous stages and events can be illustrated using a tree diagram.
Tree diagram with the three stages

Recall that there are bags, and of them contain forbidden items. The product of these numbers will give the number of the bags that contain forbidden items. The rest of the bags do not contain forbidden items. Additionally, the following two details about the bags are known.

  • If a bag contains forbidden items, there is a chance that it triggers the alarm.
  • If a bag does not have forbidden items, there is a chance that it triggers the alarm.

Considering these details, it can be concluded that of the bags containing forbidden items could trigger the alarm and of the bags that do not have forbidden items could not trigger the alarm.

Tree diagram

Now, using the percentages in the branches, the number of bags for each event can be found.

Forbidden and Alarm
Forbidden and No Alarm
Not Forbidden and Alarm
Not Forbidden and No Alarm

Finally, all the information can be shown on the tree diagram.

Tree diagram
To find the probability that a bag triggers the alarm, start by finding how many bags triggered the alarm. To do so, add the two corresponding numbers in the tree diagram. The probability that a randomly picked bag triggers the alarm is obtained dividing by the total number of bags.
b Pulling information from the tree diagram, it is seen that the number of bags that triggered the alarm and had forbidden items is and the number of bags that contained forbidden items and did not trigger the alarm is
Since the total number of bags is the ratio of the number of the favorable outcomes to the total number will give the desired probability.
Probabilities of the Events

Take note that the sum of the probabilities is equal to which is the percentage of the bags that contain forbidden items.

c To determine the probability that a bag contains forbidden items — given that it triggered the alarm — divide the total number of bags that both triggered the alarm and had forbidden items by the total number of bags that triggered the alarm.

As presented in the tree diagram, the number of bags that trigger the alarm is equal to the sum of the numbers under the word Alarm.

Therefore, there are bags that trigger the alarm. Since Mark's bag triggered the alarm, his bag is one of those Out of this group of bags, had forbidden items.
Evaluate right-hand side
Therefore, there is about a chance that Mark's bag contains forbidden items given that his bag triggered the alarm.
d To determine the probability that a bag contains forbidden items given that it did not trigger the alarm, divide the total number of bags that did not trigger the alarm and had forbidden items by the total number of bags that did not trigger the alarm.

First, find the number of bags did not trigger the alarm.

Since Izabella's bag did not trigger the alarm, her bag is one of those Out of these bags, had forbidden items.
Evaluate right-hand side
Therefore, there is about a chance that Izabella's bag contains forbidden items given that her bag did not trigger the alarm.
e If the probability of an event is closer to than then it is doubtful that the event would actually happen. With that in mind, consider the probability found in Part C.

There is about a chance that Mark's bag contains a forbidden item.

This probability is not close enough to to ensure that Mark's bag contains a forbidden item. Therefore, it is doubtful — but possible — that Mark's bag contains a forbidden item. Next, recall the answer found in Part D.

There is about a chance that Izabella's bag contains a forbidden item.

Since this probability is very small — less than — it is almost certain that Izabella does not have forbidden items in her bag — but still possible.

In this example, as well as in the previous one, dependent events and how the occurrence of one event affects the occurrence of the other event were analyzed through counting methods like tables and tree diagrams. Below, a formula for finding the probability of similar situations is shown.

Discussion

Conditional Probability

Conditional probability is the measure of the likelihood of an event occurring, given that event has occurred previously. The probability of given is written as It can be calculated by dividing the probability of the intersection of and by the probability of

It is worth noting that usually and are not equal, meaning that conditional probability is not reversible. For example, let be the event of a prime number and be the event of an odd number. The probability that a prime number is odd is almost but the reverse, an odd number being prime, is much smaller.

Why

Intuition Behind the Formula

The intuition behind the formula can be visualized by using Venn Diagrams. Consider a sample space and the events and such that

Sample Space and events A and B

Assuming that event has occurred, the sample space is reduced to

events A and B, given that A has occurred

This means that the probability that event can happen is reduced to the outcomes in the intersection of and that is, to those outcomes in

Intersection of the Events A and B

The possible outcomes are given by and the favorable outcomes by Therefore, the conditional probability formula can be obtained using the probability formula.

Now that the formula for finding a conditional probability is known, it can be put to use to solve daily life situations.

Example

Probability of Getting the Correct Size and Color

Diego's generous father has finished doing laundry and put Diego's T-shirts along with those of his big brother into the same ol’ basket. There are orange, blue, and red T-shirts in the basket, of which four are S-sized and eight are M-sized.

S-sized shirts: 1 Orange, 1 Blue, 2 Red; M-sized shirts: 5 Orange, 2 Blue, 1 Red

Diego is planning to go out with his friends. After taking a shower, he randomly picks a T-shirt from the same ol' laundry basket. Consider the following events.

a Find and compare and Write the answers as fractions in their simplest form.
b Find and compare and Write the answers as fractions in their simplest form.

Hint

a Start by finding and Then, apply the formula for finding the conditional probability.
b Find and From Part A, is already known. Apply the formula for finding conditional probability again.

Solution

a To start, remember the formula for finding the conditional probability that event happens given that event has occurred.

Applying this formula, the required probabilities can be rewritten as follows. Consequently, and need to be found. To do so, start by writing the formula to find the probability of an event. Notice that represents the probability of picking an S-size T-shirt. From the diagram, there are T-shirts in the basket, and there is only one S-size T-shirt. To find count how many of the T-shirts are From the diagram, there are only To determine count how many of the T-shirts are S-sized. From the diagram, there are only

Finding Conditional Probabilities

To find divide by
Substitute values and simplify
Therefore, the probability that Diego picks an S-size T-shirt given that he picked a orange one is or Finally, find
Substitute values and simplify
Consequently, the probability that Diego picks a T-shirt given that he picked an S-size one is or
b Similar to the previous part, start by applying the conditional probability formula to rewrite the given expressions.

From Part A, it is known that equals Thus, only and are missing. Note that is the probability of picking an S-size T-shirt. From the T-shirts, only two are and S-sized. To determine count how many of the T-shirts are From the diagram, there are

Finding Conditional Probabilities

Substituting the found probabilities into the initial equations, the required probabilities will be obtained. Start by finding
Substitute values and simplify
Therefore, the probability that Diego picks an S-size T-shirt given that he picked a blue one is or Finally, find
Substitute values and simplify
Consequently, the probability that Diego picks a T-shirt given that he picked an S-size one is or Time go to go out strutting wearing one of the shirts from the same old laundry basket — hopefully it is not his brothers!

Pop Quiz

Applying the Conditional Probability Formula

Find the required conditional probability and round it to two decimal places.

Random Probabilities of A,B, and 'A and B'

Closure

Solving the Challenge

Now that it is known how to compute conditional probabilities, Vincenzo's situation can be better investigated.
1. Tearrik next to the Eiffel Tower. 2. Tearrik in Antartica. 3. Tearrik and a penguin. 4. Frequency table comparing people who go to France/Antarctica and see a penguin with those who do not
a Vincenzo decides to travel to France. What is the probability he will see a Penguin? Round the probability to the nearest percent.
b Yesterday, Vincenzo saw a penguin! What is the probability that he was in France? Round the probability to the nearest percent.
c Vincenzo’s trip just ended. Sadly, he did not see a single Penguin. What is the probability that he chose to go to Antarctica? Round the probability to the nearest percent.

Hint

a Use the formula of conditional probability. Gathering the data from the table, a total of people participated in the survey. Of that total traveled to France and saw penguins.
b A total of people saw penguins. How many of these were in France?
c Of the people, did not see penguins and of them traveled to Antarctica.

Solution

a By the definition of conditional probability, the probability that Vincenzo saw a penguin given that he traveled to France can be expressed in the following manner.
P(Penguin|France) = P(Penguin and France)/P(France)

To find the corresponding probabilities, take a look at the table.

Frequency table comparing people who go to France/Antarctica and saw a penguin with those who did not
A total people participated in the survey and of them traveled to France and saw a penguin. Additionally, people traveled to France. Next, substitute these two probabilities into the conditional probability formula.
Substitute values and simplify
Consequently, Vincenzo has about a chance of seeing penguins, given that he traveled to France.
b In this case, the situation is opposite to that presented in Part A. Now, it is known that Vincenzo saw penguins, and it is asked to find the probability that Vincenzo is in France. As before, start by applying the conditional probability formula.
P(France|Penguin) = P(Penguin and France)/P(Penguins)
From Part A, the numerator is equal to To determine the probability of seeing penguins, determine how many of the people surveyed actually saw penguins. According to the table, people answered that they did. The next step is to substitute the two probabilities found into the conditional probability formula.
Substitute values and simplify
In conclusion, there is about a chance that Vincenzo was in France, given that he saw penguins.
c Once more, start by applying the conditional probability formula.
P(Antarctica|No Penguin) = P(Antarctica and No Penguin)/P(No Penguin)
From the second row and second column of the table, of the people traveled to Antarctica and did not see penguins. Seen in the third row of the table, people did not see penguins. Knowing this, the probability that a person picked at random did not see penguins can be computed. Finally, substitute these values into the conditional probability formula.
Substitute values and simplify
Consequently, there is about a chance that Vincenzo chose to go to Antarctica, given that he did not see penguins.
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