Question

In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form.

	Developed Flu	Did not develop Flu
Vaccine Treatment	14	1056
Placebo	95	437

Find the probability of randomly selecting 1 of the subjects and getting 1 who developed flu, given that the subject was given the vaccine treatment.

Short answer

The probability that the randomly selected subject developed flu, given that the subject received the vaccine treatment, is 0.0131.

Step-by-step solution

Given information

The counts of subjects under four different categories are tabulated.

Describe conditional probability of events 

The probability for a simple event is:

$P\left(E\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

The conditional probability of an event relative to another event that occurred in the past is expressed as$P\left(E\left|F\right.\right)$, where F is the event that is known to have occurred.

The formula for conditional probability is:

$P\left(E\left|F\right.\right)=\frac{P\left(EandF\right)}{P\left(F\right)}$

Tabulate the additive sum of columns and rows

Add the rows and columns of the counts as shown in the table.

	Developed Flu	Did not develop Flu	Totals
Vaccine Treatment	14	1056	1070
Placebo	95	437	532
Total	109	1493	1602

Express the probability of each of the events

Define event Aas choosing a subject who developed flu and B as the event of choosing a subject who received the vaccine treatment.

The number of subjects who developed flu is 109.

The number of subjects who received the vaccine treatment is 1070.

The number of subjects recorded is 1602.

The probability that the subject had developed flu is:

$P\left(A\right)=\frac{109}{1602}$

The probability that the subject had received the vaccine treatment is:

$P\left(B\right)=\frac{1070}{1602}$

The probability that the subject had developed flu and received the vaccine treatment is

$P\left(A\text{and}B\right)=\frac{14}{1602}$

Compute the conditional probability

The probability of the randomly selected subject to have developed flu, given that the subject was given the vaccine treatment, is expressed as:

$$\begin{gathered} P\left(A\left|B\right.\right)=\frac{P\left(A\text{and}B\right)}{P\left(B\right)} \\ =\frac{\frac{14}{1602}}{\frac{1070}{1602}} \\ =0.0131 \end{gathered}$$

Thus, the probability of the randomly selected subject to have developed flu, given that the subject was given the vaccine treatment, is 0.0131.