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Applications of Probability

Multiplication Rule in the Uniform Probability Model

In this lesson the Multiplication Rule of Probability will be introduced. Also, this lesson will explore how this rule can be used to find the probability of the intersection of two events and how it can be used to reverse conditional probability.

Catch-Up and Review

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

Explore

Prosecutor's Fallacy

Consider the following two events.

Event The DNA from a crime scene is concluded to be the DNA of the defendant.
Event The defendant is not guilty.

If the defendant is not guilty — event occurs — then the probability that their DNA was found on the crime scene is very small. Therefore, the conditional probability is small. Often, based on this information, the prosecutor claims that the probability that the defendant is not guilty given that occurs is also small.

This reasoning is based on the assumption that is equal to Is this assumption, and therefore the reasoning for it, correct or incorrect? Why?

Challenge

Investigating Conditional Probability

The probability of being infected with a certain disease is There is a test used to detect this disease. If someone is infected, the test comes back positive times out of cases. If someone is not infected, the test comes back positive time out of cases. Express the answers as decimal numbers approximated to one significant figure.

a What is the probability that a person is infected when they have a negative test result?
b What is the probability that a person is actually infected when they have a positive test result?
c Finally, what is the probability that a person is not infected when they have a positive test result?

Discussion

Multiplication Rule of Probability

Conversely, if and are dependent events, a rearrangement of the Conditional Probability Formula can be used to find the probability of the intersection of the events.

Proof

Consider two dependent events and The conditional probability of given is the ratio of the probability of the intersection of and to the probability of The Multiplication Rule of Probability is obtained by multiplying both sides of the above formula by and using the Symmetric Property of Equality. Following similar reasoning, an equivalent form of the rule can be obtained.


Example

Using a Tree Diagram to Determine Probability

Consider two sets of integer numbers. A number is chosen from the union of and What is the probability that the number is a positive element of What is the probability that the number is a negative element of Use a tree diagram to determine the probabilities.

Hint

Outcomes that should be included in the tree diagram are and

Solution

A tree diagram will be made to answer the questions. The root node of the tree represents the event of choosing a number from The chosen number can be either from or from

building tree diagram

The probability of each outcome should be written on the corresponding branch. Set has elements, has elements, and the sets do not have any elements in common. With this information the probability that is an element of and that is an element of can be calculated. Now the probabilities will be added to the diagram.

building tree diagram

The other outcomes that need to be considered to answer the questions are that can be positive or negative.

building tree diagram

Set has elements. of them are positive and of them are negative. Therefore, knowing that is an element of the probability that it is positive is and the probability it is negative is also

building tree diagram

Likewise, has elements: of them positive and of them negative. Knowing that is from the probability it is positive is and the probability it is negative is

tree diagram

The probability that is an element of and positive can be calculated by multiplying the probabilities along the corresponding branch of the tree diagram.

finding the probability along the branch

Similarly, the probability that is an element of and negative can be found.

finding the probability along the branch

Pop Quiz

Finding Probabilities Along Branches of a Tree Diagram

For the following questions, approximate the answers to two decimal places.

tree diagram

Example

Using a Tree Diagram to Determine Probability

Dylan participates in a school lottery that consists of two stages. He starts by drawing a ticket from a hat. This ticket tells Dylan from which of three boxes he will be drawing a second ticket. There are tickets in the hat: for box A, for box B, and for box C.

Each box has tickets. Box A has winning ticket, box B has winning tickets, and box C has What is the probability that Dylan wins? Express the answer as a percent rounded to the nearest whole number.

Hint

Use a tree diagram to represent the given information.

Solution

First make a tree diagram. The root node of the tree represents the event of drawing a ticket from the hat. There are tickets in the hat. From those, correspond to box A, correspond to box B, and correspond to box C. With this information, the probabilities for the first three branches of the tree can be written.

constructing a tree diagram

A ticket drawn from any of the boxes can be a winning or a loosing ticket. Each box contains tickets. Box A has winning ticket, box B has and box C With this information, the probabilities of drawing a winning ticket from each box can be written.

constructing a tree diagram

The sum of the probabilities of the branches that come out of the same node is equal to Knowing this, the probabilities of not drawing a winning ticket from each box can be calculated.

constructing a tree diagram

There are three possible outcomes in which Dylan draws a winning ticket.

constructing a tree diagram
The sum of the probabilities of these outcomes is the probability that Dylan draws a winning ticket.
Evaluate right-hand side

Example

Reversing Conditional Probability

Consider two boxes A and B that both contain orange and purple marbles. A box is randomly chosen and then one marble is drawn from this box. The following probabilities are known. Knowing that the drawn marble is orange, what is the probability that it was taken from box A?

Hint

How can the probability that the marble is from box A and it is orange be calculated using the given information?

Solution

The probability that a marble is from box A knowing that it is orange is a conditional probability. It is the reverse of the given conditional probability. By the definition of a conditional probability, the desired conditional probability can be expressed as follows. The probability in the denominator is known. However, the probability in the numerator is not. Since the expression in the numerator is the probability of the intersection of the events marble from box A and orange marble, it can be rewritten using the Multiplication Rule of Probability. The above expression can be substituted into the formula for the probability that the marble is from box A knowing it is orange. The conditional probability has been expressed in terms of the known probabilities. Now, the values of the known probabilities can be substituted into the formula.
Evaluate right-hand side
The probability that the marble is from box A if it is orange equals

Pop Quiz

Practice Reversing Conditional Probability

For the following questions, approximate the answers to two decimal places.

tree diagram

Closure

Investigating Conditional Probability

The following probabilities are known. Using this information, calculate three other probabilities. Write the answers correct to one significant digit.

Solution

First, the given information will be represented in a tree diagram.

tree diagram

The sum of the probabilities of branches coming out of the same node is always Using this fact, the tree diagram can be completed.

tree diagram

The tree diagram gathers all of the known information. Now, the desired unknown probabilities will be calculated one by one.

Probability That a Person Is Infected if the Test Is Negative

By the definition of conditional probability, the conditional probability that a person is infected given their test result is negative can be written as a ratio. The probability of the intersection of two events can be rewritten using the Multiplication Rule of Probability. Both probabilities in the numerator are represented in the tree diagram.

tree diagram

The probability that the test comes back negative can be found using the tree diagram as well. To do so, the probabilities of all outcomes that include receiving a negative test result should be added. There are two such outcomes. These are infected with a negative test result and not infected with a negative test result.

tree diagram
Now, the probability that a person is infected if the test is negative can be calculated.
Evaluate right-hand side

Probability That a Person Is Infected if the Test Is Positive

The probability that a person is infected if the test is positive can be written as a ratio. Again, the probability in the numerator can be rewritten using the Multiplication Rule of Probability. It is known that and Now, the probability that the test result is positive has to be found. Since the result of the test can be either positive or negative, the probability that it is positive is the difference of and the probability that the result is negative.
Finally, the desired conditional probability will be found.
Evaluate right-hand side

Probability That a Person Is Not Infected if the Test Is Positive

The probability that a person is not infected given the test result is positive can be found by reversing the conditional probability. It is known that The remaining two probabilities in the above expression are represented on the tree diagram.

tree diagram
All of the information needed to calculate the conditional probability has been found.
Evaluate right-hand side

Extra

Comparing Reversed Conditional Probabilities

The two given conditional probabilities have been reversed. Based on that, the example values of and can be compared.

It is seen that and are not equal.

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