Find the indicated probabilities using the geometric...

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Fifty-four percent of parents would give up cable television to have their child's education paid for. You randomly select five parents. Find the probability that the number of parents who would give up cable television to have their child's education paid for is (a) exactly three, (b) less than four, and (c) at least three.

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
Fifty-four percent of parents would give up cable television to have their child's education paid for. You randomly select five parents. Find the probability that the number of parents who would give up cable television to have their child's education paid for is (a) exactly three, (b) less than four, and (c) at least three.

Key Concepts

Selection of the Appropriate Probability Model

When tackling a probability problem, it is essential to choose the correct model based on the characteristics of the scenario. Problems with a fixed number of independent trials, each with a binary outcome and constant probability, suggest a binomial distribution. Alternatively, scenarios dealing with the number of trials until the first success might use the geometric distribution, while events occurring over continuous intervals at a known average rate are modeled with the Poisson distribution.

Unusual Events

An event is considered unusual if its probability is very low, typically less than 5%. Determining whether an event is unusual involves comparing the calculated probability to this benchmark, which can help in making decisions or drawing inferences about the likelihood of rare outcomes.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and the probability of success remains constant from trial to trial. It is defined by the number of trials (n) and the success probability (p), and its probability mass function determines the likelihood of exactly k successes within those n trials.

Cumulative Probabilities

Cumulative probabilities are used to determine the likelihood of a variable falling within a range rather than at a specific value. In the context of the binomial distribution, this involves summing the probabilities for a range of outcomes (such as ‘less than four’ or ‘at least three’), often utilizing tables or software to compute these cumulative values efficiently.

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