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

if 1 of the 1602 subjects is randomly selected, find the probability of getting 1 who had the vaccine treatment and developed flu.

Short answer

The probability that the subject received the vaccine treatment and developed flu is 0.00874.

Step-by-step solution

Given information

Four categories are allotted for each subject.

The categories are listed as:

• Developed flu
• Did not develop flu
• Vaccine treatment
• Placebo

Step 2:Describe the probability of any event

The probability for an event is the measure that defines the likelihood of the event.

The probability for any event is determined using the following formula:

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

Tabulate the row and column totals

Compute the sum row-wise and column-wise.

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

Express the required event mathematically

Define A as the event that the randomly selected person is vaccine treated and also develops flu.

Event A is the combination of two simple events:

• X: Subject had vaccine treatment
• Y: Subject developed flu

Thus, $P\left(A\right)=P\left(X\text{and}Y\right)$.

Compute the probability of the event

The count of the subjects in the study that had received vaccine treatment and developed flu is 14.

The total number of subjects recorded is 1602.

Using the values, the probability for event A is computed as follows:

$$\begin{gathered} P\left(A\right)=P\left(X\text{and}Y\right) \\ =\frac{14}{1602} \\ =0.00874 \end{gathered}$$

Thus, the probability that the subject in the study had received vaccine treatment and developed flu is 0.00874.