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Chapter 5: Problem 18

The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About \(10 \%\) of all adults deliberately do a one-time fling and feel no guilt about it (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press). In a group of seven adult friends, what is the probability that (a) no one has done a one-time fling? (b) at least one person has done a one-time fling? (c) no more than two people have done a one-time fling?

Short Answer

Expert verified
(a) 0.4783, (b) 0.5217, (c) 0.9999.

Step by step solution

01

Understanding the Problem

We have 7 adult friends and need to determine probabilities for their participation in a one-time fling. The probability that an individual does a one-time fling is 0.10.
02

Identify Type of Problem

This is a binomial probability problem, where the number of trials (n) is 7, and the probability of success (p) per trial is 0.10.
03

Compute Probability for Part (a)

We need the probability that no one in the group has done a one-time fling. This means x = 0, where x is the number of people having done the action. Substitute into the binomial probability formula:\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]For x = 0, we have:\[ P(0) = \binom{7}{0} (0.10)^0 (1-0.10)^7 = 1 \times 1 \times 0.9^7 \approx 0.4783 \]
04

Compute Probability for Part (b)

We want the probability that at least one person has done a one-time fling, which is the complement of no one doing a one-time fling. This can be represented as:\[ P(X \geq 1) = 1 - P(X = 0) \]Thus, we get:\[ P(X \geq 1) = 1 - 0.4783 = 0.5217 \]
05

Compute Probability for Part (c)

We need the probability that no more than two people have done a one-time fling, which means x = 0, 1, or 2. We need to calculate sum of these probabilities:\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \]First, calculate \( P(X = 1) \):\[ P(1) = \binom{7}{1} (0.10)^1 (0.90)^6 \approx 0.3729 \]Then, calculate \( P(X = 2) \):\[ P(2) = \binom{7}{2} (0.10)^2 (0.90)^5 \approx 0.1488 \]Summing these:\[ P(X \leq 2) = 0.4783 + 0.3729 + 0.1488 = 0.9999 \]
06

Conclusion

The probabilities are as follows: (a) the probability that no one has done a one-time fling is approximately 0.4783, (b) the probability that at least one person has done it is approximately 0.5217, and (c) the probability that no more than two people have done a one-time fling is approximately 0.9999.

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Key Concepts

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

Understanding Probability Theory
Probability theory is the mathematical framework that allows us to calculate the likelihood of certain events occurring. In the context of the exercise, it involves calculating the probability that a specific number of friends in a group of seven have participated in a one-time fling.

The core idea of probability is to determine the chance that an event will occur under certain conditions. The probability of an event is a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Understanding probability allows us to make predictions based on existing data or patterns.

Probability calculations can be simple, like tossing a coin, or more complex, involving multiple conditions and events, such as the current calculation involving several friends.
  • Theoretical Probability: Calculated based on possible outcomes.
  • Experimental Probability: Determined through actual experimentation or observation.
  • Probability Formula: Used to calculate the likelihood of specific outcomes, particularly useful in complex probability scenarios like the binomial distribution calculations in this exercise.
The Role of Statistical Analysis
Statistical analysis involves collecting, reviewing, and summarizing data to discover patterns and trends. In probability problems, particularly, it allows us to interpret and analyze the likelihood of outcomes systematically.

In this exercise, we're examining the occurrence of a one-time fling among a group of adult friends using statistical principles. This analysis hinges on calculating the binomial probabilities precisely.

Statistics provides tools to manage uncertainty and assess variability. Whether you're calculating the mean, median, or more advanced probabilities like the binomial distribution, statistics offers methodological guidance.
  • Descriptive Statistics: Used to summarize and describe features of a data set.
  • Inferential Statistics: Helps in making predictions or inferences about a population based on a sample. For instance, it estimates the probability that a certain number of people did a one-time fling within a given sample.
  • Probability Distributions: A function showing the possible values and likelihoods that a random variable can take. Binomial distribution, in particular, is used in cases like this exercise where we're dealing with discrete outcomes.
Exploring Binomial Distribution
The binomial distribution is a specific type of probability distribution that applies to situations where there are two possible outcomes: success or failure. This is particularly relevant in scenarios like the one-time fling problem, where each friend either has or hasn't participated in a one-time fling.

Binomials are characterized by a set number of experiments or trials, each with the same probability of success. In our problem, seven trials (friends) each have a 10% chance of having done a one-time fling. This makes it an ideal candidate for a binomial distribution.
  • Formula: The binomial probability formula is crucial as it calculates the probability of observing a specific number of successes. It is given by \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where \(n\) is the number of trials, \(x\) is the number of successes, and \(p\) is the probability of success on an individual trial.
  • Characteristics: Each trial is independent, with only two possible outcomes. This makes it versatile for many different applications in probability and statistics.
  • Applications: Beyond our textbook example, binomial distributions are applied in various fields, from finance to biology, whenever there is a need to model binary outcomes across multiple trials.

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Most popular questions from this chapter

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 \(n=2\).

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