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Chapter 4: Problem 105

Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ____ (__,___) d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl? e. Find the probability that you must ask seven people. f. Find the probability that you must ask three or four people.

Short Answer

Expert verified
a) Random variable X is the number of adults surveyed until finding one who watches. b) X = 1, 2, 3,... c) X 鈭� Geom(0.4); d) 2.5 adults; e) 0.02914; f) 0.2304.

Step by step solution

01

Define the Random Variable

The random variable \(X\) is defined as the number of adults that need to be surveyed until we find one who will watch the Super Bowl.
02

Determine Possible Values of X

\(X\) can take on any positive integer value, specifically \(X = 1, 2, 3, \ldots\). This is because theoretically, you could survey any number of people before finding one who watches the Super Bowl.
03

Identify the Distribution

Since each adult is considered independent, and the survey stops as soon as we find one who watches the Super Bowl, \(X\) follows a geometric distribution. Hence, \(X \sim \text{Geom}(0.4)\), where 0.4 is the probability of success (finding one who watches the Super Bowl).
04

Calculate the Expected Value

For a geometric distribution, the expected number of trials until the first success is given by \(E(X) = \frac{1}{p}\), where \(p\) is the probability of success. Thus, \(E(X) = \frac{1}{0.4} = 2.5\). This means we expect to survey about 2.5 adults on average.
05

Calculate Probability for Seven People

The probability that the first success occurs on the 7th trial in a geometric distribution is given by \(P(X = 7) = (1-p)^{7-1} \times p\). Substituting for our values, \(P(X = 7) = (0.6)^{6} \times 0.4 = 0.02914\).
06

Calculate Probability for Three or Four People

The probability that the first success happens at least on the 3rd or 4th survey would be \(P(X = 3) + P(X = 4)\). Calculating these individually and adding: \(P(X = 3) = (0.6)^{2} \times 0.4 = 0.144\) and \(P(X = 4) = (0.6)^{3} \times 0.4 = 0.0864\). Thus, the total probability is \(0.144 + 0.0864 = 0.2304\).

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Key Concepts

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

Probability Theory
Probability theory is the mathematical framework that allows us to quantify uncertainty. When applying it to real-world scenarios, it helps us answer questions like "What are the chances of a certain event occurring?" and "How often can I expect a certain outcome?"
In our exercise, we are focused on finding out how likely it is that the first person to watch the Super Bowl appears after a certain number of surveys. In probability terms, this event relies on an independent trial with two outcomes: either an adult watches the Super Bowl (success), or they do not (failure).
One of the fundamental properties of probability theory used in this context is the geometric distribution. This distribution is especially useful when we're interested in counting the number of trials until the first success occurs.
For a geometric distribution:
  • The probability of success, represented by \(p\), is constant for each trial.
  • The trials are independent, meaning the outcome of one survey does not impact the next.
This ensures that while many questions involve randomness or chance, probability theory gives us the tools needed to predict the likelihood of various outcomes.
Expected Value
The expected value is a key concept in probability and statistics, serving as a measure of the center of a random variable's distribution. It gives us the average outcome you can expect to see if you repeated an experiment a large number of times.
When dealing with the geometric distribution, the expected value helps us understand how many attempts, on average, we must make before experiencing the first success. This is particularly useful when planning or allocating resources for surveys or tests.
For a geometric distribution characterized by a success probability \(p\), the expected value is calculated using the formula: \[ E(X) = \frac{1}{p} \]In our example, where \(p = 0.4\), the expected number of adults to survey before finding one that watches the Super Bowl comes out to be 2.5. This indicates that, over many trials, you would survey roughly 2.5 individuals to find one who watches, though individual outcomes in any real scenario will, of course, be a whole number.
Random Variable
In probability theory, a random variable allows us to quantify outcomes of random processes numerically. It transforms real-world questions into mathematical problems. In our example, the random variable \(X\) represents the number of surveys conducted until finding an adult who watches the Super Bowl.
There are different types of random variables:
  • Discrete random variables: Take on countable values, such as the number of surveys.
  • Continuous random variables: Can take any value within a range, like measuring time or temperature, though our example focuses solely on discrete variables.
When dealing with discrete random variables, such as \(X\) in our problem, we list possible outcomes and assign probabilities to these outcomes using probability distributions (in this case, the geometric distribution).
The random variable is helpful not only for calculating probabilities of specific outcomes but also for deriving other properties of the distribution, such as its expected value, variance, or standard deviation, enabling deeper understanding and more sophisticated analysis.

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Most popular questions from this chapter

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution. Let \(X=\) the number of years a new hire will stay with the company. Let \(P(x)=\) the probability that a new hire will stay with the company \(x\) years. Complete Table 4.20 using the data provided. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 0 & {0.12} \\ \hline 1 & {0.18} \\ \hline 2 & {0.30} \\ \hline 3 & {0.15} \\ \hline 4 & {} \\\ \hline 5 & {0.10} \\ \hline 6 & {0.05} \\ \hline\end{array}$$

Identify the mistake in the probability distribution table. $$\begin{array}{|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} \\ \hline 1 & {0.15} & {0.15} \\ \hline 2 & {0.25} & {0.40} \\ \hline 3 & {0.25} & {0.85} \\\ \hline 4 & {0.20} & {0.85} \\ \hline 5 & {0.15} & {1} \\ \hline\end{array}$$

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Define the random variable \(X\).

Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {0.15} \\ \hline 2 & {0.35} \\ \hline 3 & {0.40} \\ \hline 4 & {0.10} \\ \hline\end{array}$$ On average, how many batches should the baker make?

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. Construct the probability distribution function (PDF). Stop at \(x = 6\). $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 5 & {} \\ \hline x & {P(x)} \\\ \hline 6 & {} \\ \hline\end{array}$$
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