91Ó°ÊÓ

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 \(x\) th 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 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 \(\mathrm{O}\) 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.

Step by step solution

01

Identify the known values

Identify the probability of success, which is the probability that a person is a universal donor. Given: \( p = 0.06 \). Also identify \( x \), the number of trials needed for the first success. Given: \( x = 5 \).

02

Write down the formula for the geometric distribution

The probability of getting the first success on the \( x \)th trial for a geometric distribution is given by: \[ P(x) = p (1 - p)^{x-1} \].

03

Substitute the known values into the formula

Substitute \( p = 0.06 \) and \( x = 5 \) into the formula: \[ P(5) = 0.06 (1 - 0.06)^{5-1} \].

04

Simplify the expression

First, calculate \( 1 - 0.06 \): \( 1 - 0.06 = 0.94 \). Then calculate \( 0.94^{4} \): \[ 0.94^{4} \approx 0.78 \]. Now multiply by 0.06: \[ P(5) = 0.06 \times 0.78 \approx 0.0468 \].

05

State the final probability

The probability that the first subject to be a universal blood donor is the fifth person selected is approximately 0.0468.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

probability of success

In statistics, the probability of success (denoted as \( p \)) is the chance of a particular event happening in one trial. For instance, if you're rolling a die, the probability of rolling a number greater than 3 (4, 5, or 6) is 0.5. In our exercise, we're interested in finding a universal donor. This is someone with group O and Rh-negative blood, a rare type with a probability of 0.06. The formula for a geometric distribution, which is used when identifying the first successful event in a series of otherwise independent trials, relies heavily on this probability of success. The formula is given by \( P(x) = p(1 - p)^{x-1} \), where \( x \) is the trial occurrence.

binomial distribution

The binomial distribution is used when you deal with a fixed number of trials, each with the same probability of success. Imagine flipping a coin 10 times and recording heads (H) and tails (T). Here, each flip is a trial, and the probability of success (getting heads) is 0.5 in each trial. However, if you do not fix the number of trials but look for the first success, the geometric distribution steps in. While both distributions involve repeated trials and a probability of success, they differ in the end goal. The binomial distribution counts successes in a set number of trials, and the geometric distribution seeks the first success in a possibly infinite series of trials.

National Health and Nutrition Examination Survey

The National Health and Nutrition Examination Survey (NHANES) collects various health-related data in the United States. Conducted by the National Center for Health Statistics (NCHS) and managed by the Centers for Disease Control and Prevention (CDC), NHANES helps monitor the health and nutritional statuses of the U.S. population. Participants are chosen randomly, ensuring that the data adequately represent the entire population. In our example, survey subjects are tested to identify universal donors. This population diversity makes NHANES a reliable source for understanding health and nutritional parameters across different demographics.

universal blood donor

A universal blood donor is someone with blood group O and Rh-negative type. This rare type is critical in emergencies since O negative (O-) blood can be received by any individual, regardless of their blood type. The probability of a person being a universal donor is low, typically around 0.06 or 6%. Finding the first universal donor in a sequential selection of subjects can be mathematically modeled using a geometric distribution. Each new person selected is an independent trial with a constant probability of 0.06, aiming to spot the first successful match—our universal donor.