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 0.11. Find the probability that the first subject to be a universal blood donor is the seventh person selected. The probability is (Round to four decimal places as needed.)

Solution Steps

We need to find the probability that the first subject to be a universal blood donor is the seventh person selected. The probability of being a universal donor is given as \( p = 0.11 \).

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

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

In this case, we have \( x = 7 \). Therefore, we can substitute the values into the formula:

\[ P(7) = 0.11 \cdot (1 - 0.11)^{7 - 1} \]

First, we calculate \( (1 - 0.11) \):

\[ 1 - 0.11 = 0.89 \]

Now, we raise \( 0.89 \) to the power of \( 6 \):

\[ 0.89^6 \approx 0.5277 \]

Next, we multiply this result by \( 0.11 \):

\[ P(7) = 0.11 \cdot 0.5277 \approx 0.0581 \]

Finally, we round the probability to four decimal places:

\[ P(7) \approx 0.0547 \]

Final Answer