If a procedure meets all of the conditions of a binomial distribution except 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 P(x)equalsp (1 minus p )Superscript x minus 1, where p is the probability of success on any one trial. Subjects are randomly selected for a health survey. The probability that someone is a universal donor (with group O and type Rh negative blood) is 0.05. Find the probability that the first subject to be a universal blood donor is the fourth person selected.

[tex]P(X=k) =\left[\begin{array}{ccc}n\\k\end{array}\right] p^{k}(1-p)^{n-k}[/tex] where k is the number of successes in n trials and p is the probability of success.

However, the problem tells us that when there isn't a number of trials fixed, we can use the geometric distribution and the formula for getting the first success on the xth trial becomes:

[tex]P(X=x) = p(1-p)^{x-1}\\ [/tex]

The problem asks us to find the probability of the first success on the 4th trial (given that the first subject to be a universal blood donor will be the fourth person selected)

Using this formula with the parameters given, we have:

p = 0.05

x = 4

Substituting these parameters in the formula and solving it, we get:

[tex]P(X=4) = 0.05(1-0.05)^{4-1}\\P(X=4) = 0.05 (0.95)^{3}\\P(X=4) = 0.05(.8573)\\P(X=4) = 0.0428[/tex]

Therefore, the probability that the first subject to be a universal blood donor is the fourth person selected is 0.0428 or 4.28%