Question

Geometric Distribution If a procedure meets all the conditions of a binomial distribution except that the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by , where p is the probability of success on any one trial. Subjects are randomly selected for the National Health and Nutrition Examination Survey conducted by the National Center for Health Statistics, Centers for Disease Control and Prevention. The probability that someone is a universal donor (with group O and type Rh negative blood) is 0.06. Find the probability that the first subject to be a universal blood donor is the fifth person selected.

Short answer

The probability that the first subject to be a universal blood donor is the fifth person selected is equal to 0.047.

Step-by-step solution

Given information

The probability that someone is a universal donor is given to be equal to 0.06.

Step 2:Required probability

Success is defined as selecting a person who is a universal blood donor.

Let X denote the number of trials at which success is attained, so X follows a geometric distribution.

The probability of success is defined below:

p = 0.06.

The probability of getting the first success at the xth trial is shown below:

$P\left(x\right)=p{\left(1-p\right)}^{x-1}$

Thus, the probability of the first subject to be a universal blood donor is the fifth person selected (success is attained at the fifthtrial) is computed below:

$$\begin{gathered} P\left(x\right)=p{\left(1-p\right)}^{x-1} \\ P\left(5\right)=0.06{\left(1-0.06\right)}^{5-1} \\ =0.047 \end{gathered}$$

Therefore, theprobability that the first subject to be a universal blood donor is the fifth person selected, which is equal to 0.047.