91Ó°ÊÓ

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who 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 at least eight have adequate earthquake supplies? e. Is it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why? f. How many residents do you expect will have adequate earthquake supplies?

Step by step solution

01

- Define the Random Variable X

The random variable \( X \) represents the number of California residents, out of a sample of 11, who have adequate earthquake supplies.

02

- Determine Possible Values of X

Since we are surveying 11 residents, the possible values that \( X \) can take are from 0 to 11. This means \( X \) can be 0, 1, 2, ..., or 11.

03

- Identify the Distribution

The random variable \( X \) follows a Binomial distribution where the number of trials \( n \) is 11 and the probability of success \( p \) (having adequate supplies) is 0.30. Thus, \( X \sim \text{B}(11, 0.30) \).

04

- Calculate Probability for Part D

To find the probability that at least eight have adequate supplies, calculate \( P(X \geq 8) \). This is equivalent to \( 1 - P(X \leq 7) \). We use the Binomial probability formula or a statistical calculator to find \( P(X \leq 7) \) and thus \( P(X \geq 8) = 1 - \sum_{x=0}^{7} \binom{11}{x} (0.30)^x (0.70)^{11-x} \).

05

- Compare None to All for Part E

Calculate \( P(X = 0) \) and \( P(X = 11) \) using the Binomial probability formula. Compare these values to determine whether it is more likely that none or all have adequate supplies.

06

- Determine Expected Number of Residents for Part F

The expected value for a Binomial distribution \( X \sim \text{B}(n, p) \) is \( E(X) = n \times p \). Calculate \( E(X) = 11 \times 0.30 = 3.3 \). This means, on average, about 3.3 residents are expected to have adequate earthquake supplies.

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Key Concepts

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

Random Variable

In statistical terms, a random variable is a numerical representation of a random phenomenon. It assigns numerical values to the outcomes of a random process.

In the context of our exercise, the random variable denoted by \( X \) represents the number of California residents, out of 11 surveyed, who have adequate earthquake supplies.

This means that we are interested in a numeric value indicating how many among the chosen participants actually have these supplies on hand. Understanding what this variable represents is crucial for determining probabilities and expectations within the study.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. A probability of 0 means the event cannot occur, whereas a probability of 1 indicates certainty.

For the given problem, each surveyed Californian having adequate earthquake supplies independently has a probability of 0.30, or 30%. This is our probability of success in the binomial distribution, meaning there's a 30% chance per trial (or per person) that they will have adequate supplies.

Calculating the probability of various outcomes, such as all residents having supplies or none having them, involves considering all possible numbers of 'successes' (residents with supplies) from 0 to 11.

Expected Value

Expected value in probability and statistics is a weighted average of all possible values that a random variable can take on. It gives us an idea of the 'center' or 'average' outcome we can expect over many repetitions of the experiment.

For a binomial distribution, the expected value \( E(X) \) is calculated as \( n \times p \), where \( n \) is the number of trials and \( p \) is the probability of a success on any given trial. In our exercise, \( n = 11 \) and \( p = 0.30 \), so the expected value is \( 11 \times 0.30 = 3.3 \).

This suggests that, on average, you can expect about 3.3 residents out of 11 to have adequate earthquake supplies. Although in practice, we cannot have a fraction of a person, this value guides our understanding of what a typical outcome might look like.

Statistical Calculation

When performing statistical calculations for a binomial distribution, several steps are involved to find probabilities for specific outcomes.

To determine, for example, the probability that at least 8 out of 11 residents have adequate supplies, we must calculate \( P(X \geq 8) \). This is equal to \( 1 - P(X \leq 7) \), meaning we find the probability of 7 or fewer having supplies and subtract it from 1.

The formula for binomial probability is \( \binom{n}{x} p^x (1-p)^{n-x} \), where \( \binom{n}{x} \) is the binomial coefficient and represents the number of ways to choose \( x \) successes in \( n \) trials. By calculating these probabilities, we understand the likelihood of different outcomes occurring based on our parameters.