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 2 subjects without replacement and finding that they both developed flu.

Short answer

The probability that both developed flu is 0.00459.

Step-by-step solution

Given information

The table gives the categorization of the subjects into four groups.

Describe the probability of events when carried out without replacement

The probability for a simple event is:

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

Two or more events can take place one after the other in two manners:

• With replacement: The probability of the subsequent draw remains the same.
• Without replacement: The probability of the subsequent draw changes (the previous draw is not replaced).

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 probability of each of the events

Define X as the event that the first subject developed flu and Yas the event that the second subject selected also developed flu.

The number of subjects who had developed flu is 109.

The number of remaining subjects who had developed flu after the first is selected is 108.

The total number of subjects is 1602.

The total number of remaining subjects, after the first is selected, is 1601.

The probability for events X and Y:

$$\begin{gathered} P\left(X\right)=\frac{109}{1602} \\ P\left(Y\right)=\frac{108}{1601} \end{gathered}$$

Thus, the probability that both subjects developed flu is:

$$\begin{gathered} P\left(X\text{and}Y\right)=P\left(X\right)\times P\left(Y\right) \\ =\frac{109}{1602}\times \frac{108}{1601} \\ =0.00459 \end{gathered}$$

Therefore, the probability that both subjects developed flu is 0.00459.