Smokers and Nonsmokers The participants at a mathematics...

Smokers and Nonsmokers The participants at a mathematics conference are housed dormitory-style, five to a room. Because of an oversight, conference organizers forget to ask whether the participants are smokers. In fact, it turns out that 30$\%$ are smokers. Find the probability that Fred, a nonsmoking conference participant, will be housed with: (a) Exactly one smoker. (b) One or more smokers.

Smokers and Nonsmokers The participants at a
mathematics conference are housed dormitory-style, five to a
room. Because of an oversight, conference organizers forget
to ask whether the participants are smokers. In fact, it turns
out that 30$\%$ are smokers. Find the probability that Fred, a
nonsmoking conference participant, will be housed with:
(a) Exactly one smoker.
(b) One or more smokers.

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Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is crucial for calculating the probability of a specific number of occurrences (e.g., exactly one smoker) when each trial (or room assignment) is independent and has two outcomes (smoker or nonsmoker).

Complement Principle

The complement principle is used in probability to find the likelihood of an event by subtracting the probability of its complement from one. This is particularly useful for determining the probability of one or more occurrences by first computing the probability of none, and then subtracting that value from one.

Independence

Independence in probability means that the outcome of one event does not affect the outcome of another. This concept is essential when each participant’s smoking status is determined independently, allowing the use of multiplication of probabilities for multiple independent events.

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