Some parts of California are particularly earthquake-prone....

Some parts of California are particularly earthquake-prone. Suppose that in one such area, 25$\%$ of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let $X$ denote the number among the four who have earthquake insurance. (a) Find the probability distribution of $X .[$Hint : Let $S$ denote a homeowner who has insurance and $F$ one who does not. Then one possible outcome is $S F S S,$ with probability $(.25)(.75)$ $(.25)(.25)$ and associated $X$ value $3 .$ There are 15 other outcomes. $]$ (b) Draw the corresponding probability histogram. (c) What is the most likely value for $X ?$ (d) What is the probability that at least two of the four selected have earthquake insurance?

Some parts of California are particularly earthquake-prone. Suppose that in one such area, 25$\%$ of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let $X$ denote the number among the four who have earthquake insurance.
(a) Find the probability distribution of $X .[$Hint : Let $S$ denote a homeowner who has insurance and $F$ one who does not. Then one possible outcome is $S F S S,$ with probability $(.25)(.75)$ $(.25)(.25)$ and associated $X$ value $3 .$ There are 15 other outcomes. $]$
(b) Draw the corresponding probability histogram.
(c) What is the most likely value for $X ?$
(d) What is the probability that at least two of the four selected have earthquake insurance?

Key Concepts

Complement Rule in Probability

The complement rule is a useful tool for calculating the probability of an event by subtracting the probability of its complement from 1. In this context, to calculate the probability that at least a certain number of successes occur, one might find it easier to first calculate the probability of fewer successes and then subtract that value from 1.

Mode of a Distribution

The mode of a probability distribution is the value that occurs with the highest probability, representing the most likely outcome. In the context of a binomial distribution, the mode is the value of the random variable for which the probability mass function is maximized.

Probability Histogram

A probability histogram is a graphical representation of a probability distribution, where the horizontal axis represents the possible values of the random variable and the vertical axis represents the probability of each value. It provides a visual summary of the distribution, making it easier to understand the relative likelihoods of different outcomes.

Random Variables

A random variable is a function that associates a numerical value with each outcome in a probability space. In problems like this, X is defined as the number of homeowners with earthquake insurance, making X a discrete random variable whose behavior is analyzed using probability theory.

Probability Mass Function

The probability mass function (PMF) for a discrete random variable gives the probability that the variable takes on each of its possible values. For a binomial random variable with parameters n and p, the PMF is calculated as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), which is used here to determine the probability for each possible number of insured homeowners among those selected.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. In this context, each homeowner represents an independent trial where the outcome is either having earthquake insurance (a success) or not, with a given fixed probability for each trial.

Transcript

00:01 In this question, we are told that 25 % of california homeowners have insurance against earthquakes and 75 % do not.

00:10 We randomly select four california homeowners and the random variable x is the number of the four that do have insurance against earthquakes.

00:22 So in part a, we're asked to find the probability distribution for x.

00:29 So here i've listed out all of the different possibilities for the selection of four people.

00:35 In terms of successes or failures, f stands for not having the earthquake insurance and s stands for having it.

00:44 So for the first one we have the possibility that all four have earthquake insurance, or the second one three have earthquake insurance and the fourth person does not.

00:57 And i've also worked out the probabilities for each of these individual scenarios.

01:09 So now to construct our probability distribution, we can say that the probability that x equals zero is this probability here.

01:27 That is the probability that none of the four homeowners have the insurance for earthquakes.

01:44 The probability that x is equal to one is the sum of the probabilities for any scenario that has one success.

01:59 So that is the sum of these four probabilities.

02:21 And this comes out to 0 .4219.

02:28 Similarly, the probability that x equals two is any scenario, where we have two successes and two failures.

02:41 So these are the scenarios that constitute two successes and two failures.

02:47 So what is the sum of these probabilities? and we do the same thing for x equals 3 and x equals 4.

03:30 And so this is our probability distribution for the number of homeowners out of the group of four that have insurance against earthquakes...

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