Questions: 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)=p(1-p)^(x-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 0.15. Find the probability that the first subject to be a universal blood donor is the fifth person selected.

Solution Steps

We need to find the probability that the first subject to be a universal blood donor is the fifth person selected. The probability of being a universal donor is given as \( p = 0.15 \).

The probability of getting the first success on the \( x \)-th trial is given by the formula:

\[ P(x) = p \cdot (1 - p)^{x - 1} \]

In this case, we have:

• \( p = 0.15 \)
• \( x = 5 \)

Substituting the values into the formula, we calculate:

\[ P(5) = 0.15 \cdot (1 - 0.15)^{5 - 1} \]

Calculating \( (1 - 0.15)^{4} \):

\[ (1 - 0.15)^{4} = 0.85^{4} \approx 0.52200625 \]

Now, substituting back into the equation:

\[ P(5) = 0.15 \cdot 0.52200625 \approx 0.0783009375 \]

Rounding this result to four decimal places gives:

\[ P(5) \approx 0.0783 \]

Final Answer