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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 \(x\) 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

Understanding the Problem

The probability of a person being a universal donor is given as 0.06. The task is to find the probability that the first universal donor appears on the 5th trial.

02

Formula for Geometric Distribution

The probability of getting the first success on the xth trial in a geometric distribution is given by: 饾憙(饾懃) = 饾憹(1鈭掟潙�)^{饾懃鈭�1} where P(x) is the probability, p is the success probability, and x is the trial number.

03

Substitute Given Values

In this problem, p = 0.06 and x = 5. Substitute these values into the formula: P(5) = 0.06(1 - 0.06)^{5-1}.

04

Calculate the Terms

First, calculate (1 - 0.06): 1 - 0.06 = 0.94. Next, raise 0.94 to the power of (5-1): 0.94^4 .

05

Compute the Exponent

Calculate 0.94^4: 0.94^4 鈮� 0.8154.

06

Multiply the Terms

Now, multiply 0.06 by 0.8154: P(5) 鈮� 0.06 脳 0.8154 鈮� 0.0489.

07

Conclusion

Thus, the probability that the first universal donor is the fifth person selected is approximately 0.0489.

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Key Concepts

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

headline of the respective core concept

The binomial distribution is a fundamental concept in statistics. It models the number of successes in a fixed number of independent trials. Each trial has just two possible outcomes: success or failure. The probability of success in one trial is denoted by the symbol \( p \). The binomial distribution uses the formula:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:

• \(P(X = k)\) is the probability of getting exactly \( k \) successes in \( n \) trials,
• \(\binom{n}{k}\) is the binomial coefficient (or combinations) which calculates the number of ways to choose \( k \) successes from \( n \) trials,
• \( p \) is the probability of success on a single trial, and
• \( 1-p \) is the probability of failure.

Although the geometric distribution has some different specifics, it is closely related as it calculates the probability of the first success after a series of trials given a single success probability.

headline of the respective core concept

Probability is the measure of the likelihood that an event will occur. It is calculated as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. In the geometric distribution, we are interested in the probability of getting the first success on a specific trial.

In our example, the probability of being a universal donor is \( p = 0.06 \). The probability not being a universal donor is thus \( 1 - p = 0.94 \). We want the probability that the first success (\

headline of the respective core concept

In statistics, each experiment or event is called a trial. A success trial is when the outcome of the trial meets the criteria we are examining.

For instance, in our exercise, a success trial is when a person selected out of the survey is a universal blood donor. Each trial (i.e., each person in the survey) can result in either a 'success' (they are a universal donor) or a 'failure' (they are not a universal donor). Importantly, each trial is independent, meaning the outcome of one trial does not affect the outcome of another.

headline of the respective core concept

Statistics education involves understanding and being able to interpret different measures of probability and various distributions.

Learning about the geometric distribution helps students grasp how probabilities work over sequences of trials. It's essential for analyzing real-world phenomena, such as predicting the time until a specific event occurs. For example, it is crucial in quality control, risk assessment, and many other fields.

By studying exercises like the one given, students can better understand how theoretical statistical concepts apply to practical scenarios, thereby gaining valuable insights and skills.