Questions: According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 24 flights are 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 15 flights are on time. (d) Find and interpret the probability that fewer than 15 flights are on time. (e) Find and interpret the probability that at least 15 flights are on time. (f) Find and interpret the probability that between 13 and 15 flights, inclusive, are on time.

Solution Steps

This scenario represents a binomial experiment because it satisfies the following criteria:

1. There is a fixed number of trials, \( n = 24 \) (the number of flights).
2. Each trial has two possible outcomes: success (a flight is on time) or failure (a flight is not on time).
3. The probability of success, \( p = 0.8 \), remains constant for each trial.
4. The trials are independent of each other.

The values for the binomial experiment are:

• Number of trials: \( n = 24 \)
• Probability of success: \( p = 0.8 \)

To find the probability that exactly 15 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, we find:

\[ P(X = 15) = \binom{24}{15} \cdot (0.8)^{15} \cdot (0.2)^{9} \approx 0.0236 \]

Thus, the probability that exactly 15 flights are on time is \( 0.0236 \).

In 100 trials of this experiment, it is expected that about:

\[ 100 \cdot P(X = 15) \approx 100 \cdot 0.0236 = 2.36 \]

Rounding to the nearest whole number, we expect approximately \( 2 \) trials to result in exactly 15 flights being on time.

Final Answer