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Women's Health Magazine surveyed 1187 readers to find out how often people wash their sheets. They found that even though microbiologists recommend that you wash your sheets at least once a week, only \(44 \%\) said that they wash their sheets that often. Suppose this group is representative of adult Americans and define the random variable \(x\) to be the number of adult Americans you would have to ask before you found someone that washes his or her sheets at least once a week. a. Is the probability distribution of \(x\) binomial or geometric? Explain. b. What is the probability that you would have to ask three people before finding one who washes sheets at least once a week? c. What is the probability that fewer than four people would have to be asked before finding one who washes sheets at least once a week? d. What is the probability that more than three people would have to be asked before finding one who washes sheets at least once a week?

Step by step solution

01

a. Identifying the probability distribution

In this case, we have a random variable x representing the number of adult Americans we need to ask before finding someone who washes their sheets at least once a week. This variable has only two possible outcomes: success (someone who washes their sheets at least once a week) and failure (someone who doesn't wash their sheets at least once a week). This type of distribution represents a geometric distribution, as we are dealing with the first success in a series of Bernoulli trials, with a constant probability of success at each trial.

02

b. Calculating the probability with 3 people

We need to find the probability that we have to ask three people before finding one who washes sheets at least once a week. In a geometric distribution, the probability mass function (PMF) is given by: \(P(X = k) = (1 - p)^{k-1} p\), where p is the probability of success and k is the number of trials. In our case, p = 0.44 and k = 3. So, \(P(X=3) = (1-0.44)^{3-1} * 0.44 = 0.56^2 * 0.44 \approx 0.1735\).

03

c. Calculating the probability for fewer than 4 people

We need to find the probability that fewer than four people would have to be asked before finding one who washes sheets at least once a week. This means we are looking for the probability that either 1, 2, or 3 people need to be asked. So, we can calculate the probabilities for these individual cases and then add them up: \(P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3)\) Using the geometric PMF formula from step b, we have: \(P(X < 4) = 0.44 + (0.56 * 0.44) + (0.56^2 * 0.44) \approx 0.44 + 0.2464 + 0.1735 = 0.8599\).

04

d. Calculating probability for more than 3 people

We need to find the probability that more than three people would have to be asked before finding one who washes sheets at least once a week. This is the complement of what we calculated in step c and can be written as \(P(X > 3) = 1 - P(X < 4)\). So, using the result from step c, we have: \(P(X > 3) = 1 - 0.8599 = 0.1401\).

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Key Concepts

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

Probability Distribution

Understanding the probability distribution is vital for students tackling statistical concepts. Simply put, a probability distribution is a mathematical function that describes all the possible values and likelihoods that a random variable can take within a given range. For example, when flipping a fair coin, the probability distribution would show a 50% chance for landing on heads and a 50% chance for tails.

In our exercise, we investigate how often people wash their sheets, quantified with random variable x. The probability distribution here will display different probabilities for each possible number of people you need to ask to find someone who washes their sheets weekly. Since only two outcomes are possible – they do wash (success) or they do not wash (failure) – the probability distribution for x is not uniform and will decrease as the value of x increases.

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In the context of the exercise, the random variable x represents the number of adult Americans you have to ask before finding one who washes their sheets at least once a week. A random variable is typically denoted by a capital letter, such as X, and can be discrete or continuous. Discrete random variables have a countable number of possible values, like the number of people in our example, while continuous random variables can take any value within a continuous range.

The random variable we're dealing with is discrete, as it counts the number of people, which intuitively can only be whole numbers.

Bernoulli Trials

Bernoulli trials are a series of experiments where each trial has exactly two possible outcomes, usually termed 'success' and 'failure'. These trials are independent of each other, meaning the outcome of one trial does not affect the outcome of another. The property that all trials have the same probability of success is another critical characteristic of Bernoulli trials.

In our example, each person asked about their sheet-washing habits constitutes a Bernoulli trial with two outcomes: they either wash their sheets weekly ('success') or they do not ('failure'). The survey indicates that the probability of success, which is finding someone who washes their sheets weekly, is 44%. The geometric distribution is associated with the series of Bernoulli trials that continues until the first success is observed.

Probability Mass Function

The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. Essentially, it tells us how the probability is distributed among the possible discrete values of the random variable. The PMF is associated with discrete random variables, unlike the probability density function (PDF), which is used for continuous random variables.

Returning to our sheet-washing scenario, let's apply the PMF to the geometric distribution: The PMF formula is P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success (washing sheets weekly), and k is the number of trials (people asked). Using this formula allows us to calculate probabilities for different values of k, giving us insight into scenarios like the likelihood of having to ask precisely three people before finding a weekly sheet-washer, which was part of the original exercise.