It has been estimated that only about 30% of California...

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?

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?

Key Concepts

Expected Value

For a geometric distribution, the expected (mean) number of trials until the first success is given by 1/p, where p is the probability of success on any given trial. This gives a measure of the average number of surveys needed to observe the event of interest.

Memoryless Property

One of the key characteristics of the geometric distribution is its memoryless property, meaning that the probability of success in future trials is not affected by the number of failures already observed. This property simplifies the calculation of probabilities over a sequence of independent trials.

Bernoulli Trial

Each survey of a California resident is an independent experiment with exactly two possible outcomes: either the resident does not have adequate earthquake supplies (a 'success' in the context of the experiment) or they do have adequate supplies. This binary outcome is the essence of a Bernoulli trial.

Random Variable

A random variable represents the outcome of a stochastic process. In this case, it counts the number of surveys (or trials) required until a specific event occurs—in other words, until a resident who does not have adequate earthquake supplies is found.

Geometric Distribution

The geometric distribution models the number of independent Bernoulli trials needed to obtain the first success. In this scenario, it is used to represent the number of residents surveyed until one is found who does not have adequate earthquake supplies, with its probability mass function described by P(X = k) = (1 - p)^(k-1) * p.

Transcript

00:01 In question 106, part a says, in words, define the random variable x.

00:08 So we're looking at a problem where we have an estimated 30 % of california residents have adequate earthquake supplies.

00:16 And suppose we're interested in the number of california residents.

00:19 We must survey until we find a resident who does not have adequate earthquake supplies.

00:26 So in this case, x is going to be the number of california residents.

00:33 That we survey until one does not have adequate earthquake supplies.

01:09 And b, give the list of values that x may take on.

01:13 Well, x could be one, two, and essentially it could go all the way through the population of california.

01:22 C, give the distribution of x.

01:26 Well, in this case, we're looking at surveying people until we find one who does not have adequate earthquake supplies, so this would be a geometric distribution.

01:36 And because we're told at the beginning of the problem that it's estimated that only about 30 % of california residents have earthquake supplies, that 70 % would not have adequate earthquake supplies.

01:54 D, what is the probability that we must survey just one or two residents until we find in california? resident who does not have adequate earthquake supplies.

02:04 So the probability that x is one, or, which is a plus sign here, the probability of two.

02:12 So if the first person does not have supplies, that would be 70.

02:19 If the second person, if it takes two people, then the first person was considered a failure in this case...

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