Questions: 25. Best President In a sample of 1446 U.S. registered voters, 217 said that John Kennedy was the best president since World War II. Two registered voters are selected at random without replacement. (Adapted from Quinnipiac University) (a) Find the probability that both registered voters say that John Kennedy was the best president since World War II. (b) Find the probability that neither registered voter says that John Kennedy was the best president since World War II. (c) Find the probability that at least one of the two registered voters says that John Kennedy was the best president since World War II. (d) Which of the events can be considered unusual? Explain.

Solution Steps

To find the probability that both selected voters say that John Kennedy was the best president, we use the hypergeometric distribution formula:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

Substituting the values \(N = 1446\), \(K = 217\), \(n = 2\), and \(k = 2\):

\[ P(X = 2) = \frac{\binom{217}{2} \binom{1229}{0}}{\binom{1446}{2}} = 0.0224 \]

Thus, the probability that both voters say John Kennedy was the best president is \(0.0224\).

Next, we calculate the probability that neither of the selected voters says that John Kennedy was the best president. Again, we apply the hypergeometric distribution:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

For this case, we set \(k = 0\):

\[ P(X = 0) = \frac{\binom{217}{0} \binom{1229}{2}}{\binom{1446}{2}} = 0.7223 \]

Thus, the probability that neither voter says John Kennedy was the best president is \(0.7223\).

The probability that at least one of the two voters says that John Kennedy was the best president can be calculated using the complement of the probability that neither voter says so:

\[ P(\text{at least one}) = 1 - P(X = 0) \]

Substituting the previously calculated value:

\[ P(\text{at least one}) = 1 - 0.7223 = 0.2777 \]

Thus, the probability that at least one voter says John Kennedy was the best president is \(0.2777\).

Final Answer