In a county containing a large number of rural homes, 60 %...

In a county containing a large number of rural homes, $60 \%$ of the homes are insured against fire. Four rural homeowners are chosen at random from this county, and $x$ are found to be insured against fire. Find the probability distribution for $x$. What is the probability that at least three of the four will be insured?

In a county containing a large number of rural homes, $60 \%$ of the homes are insured against fire. Four rural homeowners are chosen at random from this county, and $x$ are found to be insured against fire. Find the probability distribution for $x$.
What is the probability that at least three of the four will be insured?

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Key Concepts

Complementary and Cumulative Probabilities

Calculating the probability that at least a certain number of events occur is a common approach in probability problems. In this context, finding the probability that at least three homes are insured involves summing the probabilities for three and four successes, or using complementary probability by subtracting the probability of fewer than three successes from one.

Probability Mass Function (PMF)

The probability mass function provides the probability of exactly x successes in n independent trials in a binomial experiment. It is given by the formula P(X = x) = (n choose x) * p^x * (1-p)^(n-x), which is used to calculate the likelihood of x homes being insured out of 4.

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials where each trial has the same probability of success. It is applicable when each trial results in one of two outcomes (success or failure), and here each homeowner being insured or not represents a trial with a success probability of 0.60.

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