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 or developed flu.

Short answer

The probability that the subject had the vaccine treatment or developed flu is 0.727.

Step-by-step solution

Given information

The subjects are categorized into four groups.

Describe the probability of any event

The mathematical formula for expressing probability is:

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

The measure denotes the chance of getting an event.

Tabulate the row and column totals

The additive totals are computed for each row and column as follows:

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

Use the addition rule to solve for the probability

The addition rule for two arbitrary events Eand F states that

$P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right)$

Define A as the event that the randomly selected subject receives vaccine treatment.

Define B as the event that the randomly selected subject developed flu.

The probability that the randomly selected person either received the vaccine treatment or developed flu is expressed as $P\left(A\text{or}B\right)$.

Thus, $P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right)$ .

Compute the number of subjects in each category

The number of subjects that receive the vaccine treatment is 1070.

The number of subjects that developed flu is 109.

The number of subjects that received the vaccine treatment and developed flu is 14.

The total number of subjects recorded is 1602.

The probabilities corresponding to the events can be obtained using the table values as:

$$\begin{gathered} P\left(A\right)=\frac{1070}{1602} \\ P\left(B\right)=\frac{109}{1602} \\ P\left(A\text{and}B\right)=\frac{14}{1602} \end{gathered}$$

Substitute the values in the formula.

The probability that the randomly selected person either received the vaccine treatment or developed flu is computed as follows:

$$\begin{gathered} P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right) \\ =\frac{1070}{1602}+\frac{109}{1602}-\frac{14}{1602} \\ =0.727 \end{gathered}$$

Thus, the probability that the randomly selected person either received the vaccine treatment or developed flu is 0.727.