According to flightstats com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 20 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 12 flights are on time.
(d) Find and interpret the probability that fewer than 12 flights are on time.
(e) Find and interpret the probability that at least 12 flights are on time.
(f) Find and interpret the probability that between 10 and 12 flights, inclusive, are on time.
Answer
The values of n and p are 20 and 0.8 respectively. The probability that exactly 12 flights are on time is approximately 0.022 or 2.2%. The probability that fewer than 12 flights are on time is approximately 0.01 or 1%. The probability that at least 12 flights are on time is approximately 0.99 or 99%. The probability that between 10 and 12 flights, inclusive, are on time is approximately 0.032 or 3.2%. These probabilities can be interpreted as follows: - There is a 2.2% chance that exactly 12 out of 20 randomly selected flights from Dallas to Chicago on American Airlines are on time. - There is a 1% chance that fewer than 12 out of 20 randomly selected flights are on time. - There is a 99% chance that at least 12 out of 20 randomly selected flights are on time. - There is a 3.2% chance that between 10 and 12 out of 20 randomly selected flights, inclusive, are on time. \(\boxed{\text{Final Answer}}\)
Step 1 ：This is a binomial experiment because it meets the four conditions of a binomial experiment: 1. The experiment consists of a sequence of n identical trials (20 flights are randomly selected). 2. Two outcomes are possible on each trial (a flight is either on time or not on time). 3. The probability of success, denoted by p, is the same on each trial (the probability of a flight being on time is 80%). 4. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
Step 2 ：For part (b), n is the number of trials, which is 20 in this case. p is the probability of success on each trial, which is 0.8 (80%).
Step 3 ：For part (c), we need to find the probability that exactly 12 flights are on time. This can be calculated using the formula for the probability mass function of a binomial distribution: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where \(C(n, k)\) is the binomial coefficient "n choose k", p is the probability of success, and n is the number of trials.
Step 4 ：For part (d), we need to find the probability that fewer than 12 flights are on time. This is the sum of the probabilities that 0, 1, 2, ..., 11 flights are on time.
Step 5 ：For part (e), we need to find the probability that at least 12 flights are on time. This is 1 minus the probability that fewer than 12 flights are on time.
Step 6 ：For part (f), we need to find the probability that between 10 and 12 flights, inclusive, are on time. This is the sum of the probabilities that 10, 11, and 12 flights are on time.
Step 7 ：The values of n and p are 20 and 0.8 respectively. The probability that exactly 12 flights are on time is approximately 0.022 or 2.2%. The probability that fewer than 12 flights are on time is approximately 0.01 or 1%. The probability that at least 12 flights are on time is approximately 0.99 or 99%. The probability that between 10 and 12 flights, inclusive, are on time is approximately 0.032 or 3.2%. These probabilities can be interpreted as follows: - There is a 2.2% chance that exactly 12 out of 20 randomly selected flights from Dallas to Chicago on American Airlines are on time. - There is a 1% chance that fewer than 12 out of 20 randomly selected flights are on time. - There is a 99% chance that at least 12 out of 20 randomly selected flights are on time. - There is a 3.2% chance that between 10 and 12 out of 20 randomly selected flights, inclusive, are on time. \(\boxed{\text{Final Answer}}\)
