Worst President In a sample of 2016 U.S. adults, 242 said...

Worst President In a sample of 2016 U.S. adults, 242 said Richard Nixon was the worst president since World War II. Three U.S. adults are selected at random without replacement. (Adapted from Harris Interactive) (a) Find the probability that all three adults say Richard Nixon was the worst president since World War II. (b) Find the probability that none of the three adults say Richard Nixon was the worst president since World War II. (c) Find the probability that at most two of the three adults say Richard Nixon was the worst president since World War II. (d) Which of the events can be considered unusual? Explain.

Worst President In a sample of 2016 U.S. adults, 242 said Richard Nixon was the worst president since World War II. Three U.S. adults are selected at random without replacement. (Adapted from Harris Interactive)
(a) Find the probability that all three adults say Richard Nixon was the worst president since World War II.
(b) Find the probability that none of the three adults say Richard Nixon was the worst president since World War II.
(c) Find the probability that at most two of the three adults say Richard Nixon was the worst president since World War II.
(d) Which of the events can be considered unusual? Explain.

Key Concepts

Complement Rule in Probability

The complement rule is used to simplify probability calculations by finding the probability of an event not occurring and then subtracting from one. It is particularly useful when calculating probabilities involving 'at most' or 'at least' type events.

Sampling Without Replacement

Sampling without replacement means that once an individual item is selected from the population, it is not returned to the pool for possible re-selection. This affects probabilities on subsequent draws because the population size and the number of successes left are both reduced.

Definition of Unusual Events

In probability and statistics, an event is typically considered unusual if its probability is very low, often below a conventional threshold such as 5%. Understanding this concept helps in deciding whether an observed outcome is likely to be a result of random chance or if it merits further investigation.

Hypergeometric Distribution

This is a probability distribution used for sampling without replacement from a finite population. It is applicable when there are only two outcomes (such as 'success' or 'failure') in each draw, and it calculates the likelihood of drawing a certain number of successes in a given sample size.

Combinatorics

This area of mathematics deals with counting, combinations, and permutations. In probability problems, especially those involving selections from a finite set, combinatorial methods are used to count the number of favorable outcomes and the total number of possible outcomes, allowing for the calculation of probabilities.

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