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It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose we are interested in the number of California residents we must survey until we find a resident who does not have adequate earthquake supplies. 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. What is the probability that we must survey just one or two residents until we find a California resident who does not have adequate earthquake supplies? e. What is the probability that we must survey at least three California residents until we find a California resident who does not have adequate earthquake supplies? f. How many California residents do you expect to need to survey until you find a California resident who does not have adequate earthquake supplies? g. How many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies?

Step by step solution

01

Define the Random Variable

In this problem, we define the random variable \( X \) as the number of California residents surveyed until we find a resident who does not have adequate earthquake supplies.

02

List Values for the Random Variable

Since \( X \) counts the number of trials until the first success (finding a resident without adequate supplies), it can take on the values \( 1, 2, 3, \ldots \). These are all positive integers.

03

Identify the Distribution for X

Given that \( X \) represents the number of trials until the first success in a series of Bernoulli trials (each with the same probability of success), \( X \) follows a geometric distribution characterized by the parameter \( p \), which is the probability of success (not having adequate supplies). Since 30% do have adequate supplies, 70% do not, so the probability \( p = 0.7 \). Thus, \( X \sim \text{Geom}(0.7) \).

04

Calculate Probability for One or Two Surveys

We need to find the probability that \( X = 1 \) or \( X = 2 \). The probability mass function for a geometric distribution is given by \( P(X = k) = (1 - p)^{k-1} \cdot p \). Therefore: - \( P(X = 1) = p = 0.7 \)- \( P(X = 2) = (1 - p) \cdot p = 0.3 \times 0.7 = 0.21 \)Thus, the probability is \( 0.7 + 0.21 = 0.91 \).

05

Calculate Probability for At Least Three Surveys

To find \( P(X \geq 3) \), compute \( 1 - P(X = 1) - P(X = 2) \). From Step 4, \( P(X = 1) = 0.7 \) and \( P(X = 2) = 0.21 \). Therefore, \[ P(X \geq 3) = 1 - 0.7 - 0.21 = 0.09 \]

06

Expected Value for Residents Without Supplies

The expected value \( E(X) \) for a geometric distribution is given by \( \frac{1}{p} \). With \( p = 0.7 \), we find \[ E(X) = \frac{1}{0.7} \approx 1.43 \] residents are expected to be surveyed to find one without adequate supplies.

07

Expected Value for Residents With Supplies

Since 30% of residents have adequate supplies, let \( Y \) be the number of residents surveyed until we find one who does have adequate supplies, also following a geometric distribution with \( q = 0.3 \). The expected value is \[ E(Y) = \frac{1}{q} = \frac{1}{0.3} \approx 3.33 \] residents surveyed to find one with adequate supplies.

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Key Concepts

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

Random Variable

In probability and statistics, a random variable is a variable that takes on different numerical values based on the outcomes of a random phenomenon. In the context of the exercise, the random variable is denoted by \(X\). What \(X\) represents in this scenario is the number of California residents we must survey until we identify a resident who does not have adequate earthquake supplies.
This means \(X\) can take any positive integer value, such as 1, 2, 3, and so on, because we might need to survey 1 person or more until we find one who lacks supplies.
Understanding what a random variable represents is crucial as it helps in determining the type of distribution it follows, and forms the basis for further statistical analysis.

Bernoulli Trials

A Bernoulli trial is a random experiment that results in a binary outcome, typically labeled 'success' and 'failure'. Each trial is independent of the others, meaning the outcome of one trial does not affect the subsequent trials. In the scenario given, each survey of a resident can be considered a Bernoulli trial.
Here, a 'success' is defined as finding a resident who does not have adequate earthquake supplies, with a probability \(p = 0.7\). This probability is derived from the information that 70% of residents do not have these supplies. This setup is a classic example of Bernoulli trials, where we continue the trials until the first success occurs.
Recognizing whether a problem involves Bernoulli trials helps one determine the appropriate mathematical model to use, in this case, the geometric distribution.

Probability Mass Function

The probability mass function (PMF) specifies the probability that a discrete random variable is exactly equal to some value. For the geometric distribution, which our exercise follows, the PMF is expressed as:
\[ P(X = k) = (1 - p)^{k-1} \ p \]
where \(k\) is the number of trials, and \(p\) is the probability of success for each trial. For instance, if we want to calculate the probability that exactly one or two residents must be surveyed before finding one without supplies, we utilize this PMF function.
- For \(k = 1\), \(P(X = 1) = 0.7\).- For \(k = 2\), \(P(X = 2) = 0.3 \ times \ 0.7 = 0.21\).
Adding these gives the total probability of needing one or two surveys. The PMF provides a precise method for calculating these probabilities and offers insight into the distribution's behavior.

Expected Value

Expected value is a fundamental concept in probability representing the average outcome if an experiment is repeated a large number of times. For a geometric distribution, the expected value can be easily computed using the formula \( E(X) = \frac{1}{p} \), where \(p\) is the probability of success in a single trial.
In our context, the probability of success (finding a resident without adequate supplies) is \(p = 0.7\). Therefore, the expected value is:
\[ E(X) = \frac{1}{0.7} \approx 1.43 \]
This suggests, on average, you would need to survey approximately 1.43 residents to find one without adequate earthquake supplies.
Understanding expected value helps in predicting outcomes over a long run of data, giving insights into what one might realistically expect in numerous trials.