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Applications of Probability

Conditional Probability in the Uniform Probability Model

In this lesson, a variety of conditional situations will be modeled by using tree diagrams, frequency tables, and Venn diagrams.

Catch-Up and Review

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

Challenge

Detecting if an Email Is Spam

Spam filters determine whether an email is spam by checking it for some words that appear more frequently in spam emails. The following set of information is known.

  • of emails are spam.
  • of spam emails contain the word free.
  • of non-spam emails contain the word free.

Jordan gets an email with the word free in it.

a Draw a tree diagram to visualize the situation.
b What is the probability that the email is spam?

Discussion

Reviewing the Intuition Behind the Conditional Probability Formula

Recall the formula for the conditional probability.

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

It means that the probability that event can happen is reduced to the outcomes in the intersection of events and or

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.

Example

Using a Venn Diagram to Find Conditional Probability

Dominika and her friends, people in total, want to play basketball. They decide to form two teams randomly. To do so, each draws a card from a stack of cards numbered from to

a Dominika excitedly draws the first card. What is the probability that she draws the number Write the probability as a fraction in its simplest form.
b If Dominika is on Team Red, what is the probability that she drew the number Write the probability as a fraction in its simplest form.

Hint

a Determine the sample space.
b Start by drawing a Venn diagram to represent the situation. Use the diagram to determine the sample space of the situation.

Solution

a Since Dominika is the first one to draw a card, there are options for her. Therefore, the sample space for this case consists of numbers from to

The favorable outcome is to draw the number Using the Probability Formula, the probability of drawing is the ratio of the number of favorable outcomes to the total number of outcomes

b The given situation will be represented by using a Venn diagram. The following two events will be examined.

The rest of the possible outcomes will be shown outside of the events and

Sample Space and events A and B

Since Dominika is on Team Red, the sample space is reduced to the outcomes in

Sample space reduced to outcomes in A

There are odd numbers in the new sample space and only favorable outcome.

Favorable outcome in A

Therefore, using the Probability Formula, the probability that Dominika drew the number given that the number drawn is odd, is found as follows.

Pop Quiz

Practice Finding Conditional Probability

The applet shows the probabilities of two events in a Venn diagram. Calculate the conditional probabilities. If necessary, round the answer to two decimal places.

Probabilities of events A and B shown in Venn diagram
It is practical to model many real-life situations with a uniform probability model, even though they may not actually be uniform after close inspection. Examine the next example and interpret the results.

Example

Using a Table to Find Conditional Probabilities

After reading an article about the famous wreck of the Titanic, Paulina concluded that the rescue procedures favored the wealthier first-class passengers. She then finds some data on the survival of the Titanic passengers.

Survived Did Not Survive Total
First Class Passengers
Second Class Passengers
Third Class Passengers
Total

Use this data to investigate the probabilities of surviving the wreck of the Titanic.

a Determine if the events are independent or not. Justify your answer using appropriate probability calculations.
Passenger Survived
First Class Passenger
Second Class Passenger
Third Class Passenger
b Did all passengers aboard the Titanic have the same probability of surviving? Justify your answer.

Answer

a Table:
Passenger Survived
First Class Passenger Dependent,
Second Class Passenger Dependent,
Third Class Passenger Dependent,
b No, see solution.

Hint

a Use the fact that two events and are independent if
b Compare the conditional probabilities and

Solution

a If two events are independent, the occurrence of one of the events does not affect the occurrence of the other. In other words, the probability that occurs given that has already occurred is the same as the probability that occurs.
Therefore, the following probabilities need to be calculated first. To do so, the Probability Formula will be used. Now, examine the given table to find the probabilities of each event. Adding the event labels to the table can make finding the probabilities a little easier. Recall that when finding a conditional probability, the sample space is reduced.
Using table to find the probabilities
Having found the probabilities, note that none of the conditional probabilities is equal to
Fraction Decimal

Therefore, events and each have an effect on event meaning that is a dependent event. In simpler terms, a passenger's chance of surviving depended on what class they were traveling in.

Passenger Survived
First Class Passenger Dependent,
Second Class Passenger Dependent,
Third Class Passenger Dependent,
b In the previous part, the conditional probabilities and were found.

Comparing these probabilities, it can be concluded that not all passengers aboard the Titanic had the same chance of surviving. The first class passengers had the greatest chance of being rescued.

It would be appropriate to assume that each passenger had the same chance of survival, as the uniform probability model suggests. However, the data shows that not all passengers had the same chance of survival.

Pop Quiz

Practice Finding Conditional Probability

The applet shows the frequency of each event in a table. Calculate the conditional probability asked in the applet. If necessary, round the answer to two decimal places.

Table showing the frequencies of events

Example

Using a Tree Diagram to Find Conditional Probabilities

Tadeo searches the Internet to check how effective Drug A is compared to Drug B. He finds a research paper about the drugs that gives the following information.

  • One third of the participants received Drug A, one third received Drug B, and one third received a placebo.
  • of participants who received Drug A, of participants who Drug B, and of participants who received the placebo reported recovery from their condition.

Help Tadeo answer the following questions.

a Draw a tree diagram to represent the situation.
b What is the probability that a participant did not recover if they received Drug B?
c Suppose that the number of participants is What is the number of participants that received Drug A and did not recover?

Answer

a Example Tree Diagram:
Tree diagram of the situation
b
c

Hint

a Start by considering the first point of the information Tadeo found. How many outcomes are there initially?
b Determine if it is a conditional probability or not. Use the tree diagram to find it.
c Start by finding the probability of a participant receiving Drug A and not recovering.

Solution

a Consider the first point of the information found by Tadeo.
  • One third of the participants received Drug A, one third recieved Drug B, and one third received a placebo.

Since there are three possible outcomes, start by drawing three branches and labeling them. Additionally, since the drugs were distributed among participants evenly, each branch has the same probability,

First part of tree diagram

Each of these outcomes will have two further outcomes — recovered or not recovered Therefore, two more branches will be drawn for each case.

Second part of tree diagram

Each branch should have a probability value on it. Notice that these probabilities are conditional probabilities.

Conditional probabilities on tree diagram

Since there are only two possible outcomes after a certain medicine is received, these outcomes are complements of each other. Therefore, by the Complement Rule, the following equations can be written. Next, these conditional probabilities can be calculated using the information from the second point.

  • of participants who received Drug A, of participants who Drug B, and of participants who received the placebo reported recovery from their condition.
Knowing that of the participants received Drug A and or of the participants who received Drug A recovered, can be calculated using the Conditional Probability Formula.
Simplify right-hand side
When a participant receives Drug A, of them recover, which means that do not recover. The other conditional probabilities can be found in a similar fashion. The percentages will be written as decimals when calculating each probability. The other conditional probabilities can be found in the table.
Conditional Probability Formula Substitute Evaluate Probability of the Complement

Finally, the tree diagram can be completed.

Tree diagram
b A participant's well-being varies depending on the drug they received. The probability that a participant did not recover if they received Drug B is a conditional probability because knowing that a participant received Drug B changes the probability of recovery.

Recall that each final branch of the tree diagram represents a conditional probability. The branch between and will represent the probability of a participant receiving Drug B and not recovering.

Highlighted branch between B and NR

Therefore, the conditional probability that a participant did not recover if they received Drug B is

c Consider the path through to
Path through A to NR
The highlighted path represents the probability that a participant received Drug A and did not recover. This probability is equal to the product of the probabilities on the branches. The number of study participants who received Drug A and did not recover can be found by multiplying this probability by the total number of participants, people.
Simplify
participants received Drug A and did not recover from their condition.

Closure

Detecting if an Email Is Spam

Like Venn diagrams and frequency tables, tree diagrams relate the probability of a conditional event to a subset of the event occurring. With this in mind, reconsider the example given at the beginning of the lesson. These three points about spam emails are known.

  • of emails are spam.
  • of spam emails contain the word free.
  • of non-spam emails contain the word free.

Jordan gets an email with the word free in it.

a Draw a tree diagram to visualize the situation.
b What is the probability that the email is spam? If necessary, round the answer to the two decimal places.

Answer

a Example Tree Diagram:
Tree diagram visualizing the spam email situation
b

Hint

a Start by finding the probabilities that will be written on branches.
b Determine the paths of the tree diagram that lead to emails that contains the word free.

Solution

a Start by defining the events.
Denoting each event with symbols

Therefore, and are given. Notice that is the complement of and is the complement of Recall that the probability of the complement of an event is minus the probability of the event. Therefore, the probabilities of the complements of the given events can be found.

With the events and probabilities determined, a tree diagram can be drawn as shown.

Tree diagram visualizing the spam email situation
b Jordan gets an email that contains the word free. The probability that this email is spam needs to be calculated.

To find it, the tree diagram drawn in the previous part will be used. Consider all the paths that lead to event the event that an email contains the word free.

Tree diagram visualizing the spam email situation
The probability of either of these two paths occurring is the sum of the products of the probabilities on the paths. Of these two paths, the topmost one leads to the favorable event. The probability of this event is the product of the probabilities on the path. As a result, the ratio of the probability of the favorable path to the probability of either of the possible paths gives
An email that contains the word free has a probability that it is a spam email.
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