Questions: According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 17 flights are randomly selected, and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Determine the values of n and p. (c) Find and interpret the probability that exactly 11 flights are on time. (d) Find and interpret the probability that fewer than 11 flights are on time. (e) Find and interpret the probability that at least 11 flights are on time. (f) Find and interpret the probability that between 9 and 11 flights, inclusive, are on time.

Solution Steps

This scenario qualifies as a binomial experiment due to the following reasons:

• There are two mutually exclusive outcomes: success (the plane arrives on time) or failure (the plane does not arrive on time).
• The trials are independent, meaning the outcome of one flight does not affect the others.
• The probability of success remains constant at \( p = 0.8 \) for each flight.
• The experiment is conducted a fixed number of times, specifically \( n = 17 \) flights.

The values for the binomial experiment are:

• Number of trials \( n = 17 \)
• Probability of success \( p = 0.8 \)

To find the probability that exactly 11 flights are on time, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Substituting the values:

\[ P(X = 11) = \binom{17}{11} \cdot (0.8)^{11} \cdot (0.2)^{6} \approx 0.068 \]

Thus, the probability that exactly 11 flights are on time is approximately \( 0.068 \).

Final Answer