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Chapter 5: Problem 6

Basic Computation: Geometric Distribution Given a binomial experiment with probability of success on a single trial \(p=0.30,\) find the probability that the first success occurs on trial number \(n=2\).

Short Answer

Expert verified
The probability is 0.21.

Step by step solution

01

Understanding the Geometric Distribution

A geometric distribution models the number of trials until the first success in a series of independent and identically distributed Bernoulli trials. For this problem, the probability of success on any given trial is given by \( p = 0.30 \).
02

Formula Setup for Geometric Probability

The probability that the first success occurs on the \( n \)-th trial is given by \( P(X = n) = (1-p)^{n-1} \cdot p \). This is because the first \( n-1 \) trials must be failures and the \( n \)-th trial must be a success.
03

Plugging in Values

For \( n = 2 \), we plug the values into the formula: \( P(X = 2) = (1-0.30)^{2-1} \cdot 0.30 \).
04

Calculating Probability of Failure and Success

First, calculate \( (1-0.30) = 0.70 \). Then, substitute \( n-1 = 1 \) into the equation: \( (0.70)^1 = 0.70 \). Finally, multiply by the probability of success: \( 0.70 \times 0.30 \).
05

Final Calculation

Calculate the final result: \( 0.70 \cdot 0.30 = 0.21 \). This is the probability that the first success occurs on trial number 2.

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Key Concepts

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

Binomial Experiment
A binomial experiment is a statistical test that involves performing a series of trials, where each trial results in one of two possible outcomes鈥攕uccess or failure. These trials are repeated under the same conditions. The number of successes in the trials follows a binomial distribution.
  • In a binomial experiment, the probability of success (\(p\)) stays constant from one trial to the next.
  • The trials are independent, meaning the outcome of one trial does not influence the others.
Let's imagine flipping a coin, where each flip is a trial. We often use binomial experiments in statistics to determine the likelihood of a certain number of successes over a series of trials. The geometric distribution, which concerns the number of trials until the first success, relies on this setup but emphasizes the sequence rather than the total number of successes.
Probability of Success
The probability of success in a single trial is a crucial concept in both binomial and geometric distributions. It is a fixed number that represents how likely a successful outcome is in one attempt.
  • For example, in a coin flip, the probability of landing heads (a success) might be 0.50.
  • In our example problem, the probability of success on any given trial was 0.30.
In the context of our problem, a success is defined in terms of the variable being measured鈥攊n this exercise, the success was achieving a successful trial on the second attempt. Knowing the probability of success allows us to calculate the likelihood of different outcomes using the formula for the geometric distribution: \[P(X = n) = (1-p)^{n-1} \cdot p\]Here, \(p\) represents the probability of success.
Bernoulli Trials
Bernoulli trials are the individual experiments conducted in the framework of a binomial experiment. Named after the Swiss mathematician Jacob Bernoulli, each trial has exactly two possible outcomes. In many practical scenarios, they form the foundation for understanding more complex statistical models.
  • Each Bernoulli trial is independent.
  • The outcome of any single trial does not affect subsequent trials.
  • The probability of each specific outcome remains constant throughout the trials.
These trials are a key element in deriving the geometric distribution, as they define the mathematical environment for calculating when the first success will occur. Each trial's outcome being independent and identical makes it perfect for describing real-world processes where a single outcome is of interest over several attempts.

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Most popular questions from this chapter

Jim is a 60 -year-old Anglo male in reasonably good health. He wants to take out a 50,000 dollar term (i.e., straight death benefit) life insurance policy until he is \(65 .\) The policy will expire on his 65 th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th edition). $$\begin{array}{|l|ccccc|} \hline x=\text { age } & 60 & 61 & 62 & 63 & 64 \\\ \hline P( \text { death at this age) } & 0.01191 & 0.01292 & 0.01396 & 0.01503 & 0.01613 \\ \hline \end{array}$$ Jim is applying to Big Rock Insurance Company for his term insurance policy. (a) What is the probability that Jim will die in his 60 th year? Using this probability and the 50,000 dollar death benefit, what is the expected cost to Big Rock Insurance? (b) Repeat part (a) for years \(61,62,63,\) and \(64 .\) What would be the total expected cost to Big Rock Insurance over the years 60 through \(64 ?\) (c) If Big Rock Insurance wants to make a profit of 700 dollar above the expected total cost paid out for Jim's death, how much should it charge for the policy? (d) If Big Rock Insurance Company charges 5000 dollar for the policy, how much profit does the company expect to make?

What does the random variable for a binomial experiment of \(n\) trials measure?

In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

What is the age distribution of patients who make office visits to a doctor or nurse? The following table is based on information taken from the Medical Practice Characteristics section of the Statistical Abstract of the United States (116th edition). $$\begin{array}{l|ccccc} \hline \text { Age group, years } & \text { Under 15 } & 15-24 & 25-44 & 45-64 & 65 \text { and older } \\ \hline \text { Percent of office visitors } & 20 \% & 10 \% & 25 \% & 20 \% & 25 \% \\ \hline \end{array}$$ Suppose you are a district manager of a health management organization (HMO) that is monitoring the office of a local doctor or nurse in general family practice. This morning the office you are monitoring has eight office visits on the schedule. What is the probability that (a) at least half the patients are under 15 years old? First, explain how this can be modeled as a binomial distribution with 8 trials, where success is visitor age is under 15 years old and the probability of success is \(20 \%\) (b) from 2 to 5 patients are 65 years old or older (include 2 and 5 )? (c) from 2 to 5 paticnts are 45 years old or older (include 2 and 5 )? Hint: Success is 45 or older. Use the table to compute the probability of success on a single trial. (d) all the patients are under 25 years of age? (e) all the patients are 15 years old or older?

USA Today reported that the U.S. (annual) birthrate is about 16 per 1000 people, and the death rate is about 8 per 1000 people. (a) Explain why the Poisson probability distribution would be a good choice for the random variable \(r=\) number of births (or deaths) for a community of a given population size. (b) In a community of 1000 people, what is the (annual) probability of 10 births? What is the probability of 10 deaths? What is the probability of 16 births? 16 deaths? (c) Repeat part (b) for a community of 1500 people. You will need to use a calculator to compute \(P(10 \text { births) and } P(16\text { births). }\) (d) Repeat part (b) for a community of 750 people.
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