Questions: Your flight has been delayed: At Denver International Airport, 81% of recent flights have arrived on time. A sample of 12 flights is studied. Round the probabilities to at least four decimal places. Part: 0 / 4 Part 1 of 4 (a) Find the probability that all 12 of the flights were on time. The probability that all 12 of the flights were on time is .

Solution Steps

We are tasked with finding the probability that all 12 flights arrive on time at Denver International Airport, where the probability of a flight arriving on time is \( p = 0.81 \).

The probability of exactly \( x \) successes (flights on time) in \( n \) trials (flights) can be calculated using the binomial probability formula:

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

where:

• \( n = 12 \) (total flights),
• \( x = 12 \) (flights on time),
• \( p = 0.81 \) (probability of a flight being on time),
• \( q = 1 - p = 0.19 \) (probability of a flight not being on time).

Substituting the values into the formula, we have:

\[ P(X = 12) = \binom{12}{12} \cdot (0.81)^{12} \cdot (0.19)^{0} \]

Calculating each component:

• \( \binom{12}{12} = 1 \)
• \( (0.81)^{12} \approx 0.0798 \)
• \( (0.19)^{0} = 1 \)

Thus, the probability simplifies to:

\[ P(X = 12) = 1 \cdot 0.0798 \cdot 1 = 0.0798 \]

Final Answer