Understanding the Multiplication Rule of Probability

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

Event

A

The DNA from a crime scene is concluded to be the DNA of the defendant.

Event

B

The defendant is not guilty.

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

The sum of the probabilities of branches coming out of the same node is always $1 .$ Using this fact, the tree diagram can be completed.

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.

P (infected ∣ negative) = \frac{P ( infected \cap negative )}{P ( negative )}

The probability of the intersection of two events can be rewritten using the Multiplication Rule of Probability.

P (infected ∣ negative) = \frac{P ( negative ∣ infected ) \cdot P ( infected )}{P ( negative )}

Both probabilities in the numerator are represented in the 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.

Now, the probability that a person is infected if the test is negative can be calculated.

P (infected ∣ negative) = \frac{P ( negative ∣ infected ) \cdot P ( infected )}{P ( negative )}

SubstituteValues

Substitute values

P (infected ∣ negative) = \frac{0 . 0 3 \cdot 0 . 0 0 0 1}{0 . 9 9 9 8 0 3 0 1}

Evaluate right-hand side

Multiply

P (infected ∣ negative) = \frac{0 . 0 0 0 0 0 3}{0 . 9 9 9 8 0 3 0 1}

UseCalc

Use a calculator

P (infected ∣ negative) = 0.0000030005 \dots

RoundSigDig

Round to $1$ significant digit(s)

P (infected ∣ negative) \approx 0.000003

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.

P (infected ∣ positive) = \frac{P ( infected \cap positive )}{P ( positive )}

Again, the probability in the numerator can be rewritten using the Multiplication Rule of Probability.

P (infected ∣ positive) = \frac{P ( positive ∣ infected ) \cdot P ( infected )}{P ( positive )}

It is known that

P (infected) = 0.0001

and

P (positive ∣ infected) = 0.97 .

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

1

and the probability that the result is negative.

P (positive) = 1 - P (negative)

Substitute

$P (negative) = 0.99980301$

P (positive) = 1 - 0.99980301

SubTerm

Subtract term

P (positive) = 0.00019699

Finally, the desired conditional probability will be found.

P (infected ∣ positive) = \frac{P ( positive ∣ infected ) \cdot P ( infected )}{P ( positive )}

SubstituteValues

Substitute values

P (infected ∣ positive) = \frac{0 . 9 7 \cdot 0 . 0 0 0 1}{0 . 0 0 0 1 9 6 9 9}

Evaluate right-hand side

Multiply

P (infected ∣ positive) = \frac{0 . 0 0 0 0 9 7}{0 . 0 0 0 1 9 6 9 9}

UseCalc

Use a calculator

P (infected ∣ positive) = 0.49241 \dots

RoundSigDig

Round to $1$ significant digit(s)

P (infected ∣ positive) \approx 0.5

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.

P (not infected ∣ positive) = \frac{P ( positive ∣ not infected ) \cdot P ( not infected )}{P ( positive )}

It is known that

P (positive) = 0.00019699 .

The remaining two probabilities in the above expression are represented on the tree diagram.

All of the information needed to calculate the conditional probability has been found.

P (not infected ∣ positive) = \frac{P ( positive ∣ not infected ) \cdot P ( not infected )}{P ( positive )}

SubstituteValues

Substitute values

P (not infected ∣ positive) = \frac{0 . 0 0 0 1 \cdot 0 . 9 9 9 9}{0 . 0 0 0 1 9 6 9 9}

Evaluate right-hand side

Multiply

P (not infected ∣ positive) = \frac{0 . 0 0 0 0 9 9 9 9}{0 . 0 0 0 1 9 6 9 9}

UseCalc

Use a calculator

P (not infected ∣ positive) = 0.50758 \dots

RoundSigDig

Round to $1$ significant digit(s)

P (not infected ∣ positive) \approx 0.5

$P (positive \| infected)$	$P (infected \| positive)$
$0.97$	$0.5$
$P (positive \| not infected)$	$P (not infected \| positive)$
$0.0001$	$0.5$

P (positive | infected)

P (infected | positive)

0.97

0.5

P (positive | not infected)

P (not infected | positive)

0.0001

0.5

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Catch-Up and Review

Explore

Prosecutor's Fallacy

Challenge

Investigating Conditional Probability

Discussion

Multiplication Rule of Probability

Proof

Example

Using a Tree Diagram to Determine Probability

Hint

Solution

Pop Quiz

Finding Probabilities Along Branches of a Tree Diagram

Example

Using a Tree Diagram to Determine Probability

Hint

Solution

Example

Reversing Conditional Probability

Hint

Solution

Pop Quiz

Practice Reversing Conditional Probability

Closure

Investigating Conditional Probability

Solution

Probability That a Person Is Infected if the Test Is Negative

Probability That a Person Is Infected if the Test Is Positive

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

Extra

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