The problem involves calculating probabilities for a binomial distribution, as we are given a fixed number of trials (7 flights), with two possible outcomes (on time or not on time), and a constant probability of success (50% or 0.5 for an on-time flight).

Let:

• $n = 7$ (number of trials or flights),
• $p = 0.5$ (probability of success, i.e., flight being on time),
• $q = 1 - p = 0.5$ (probability of failure, i.e., flight not being on time).

The binomial probability formula is: $P(X = k) = \binom{n}{k} p^k q^{n-k}$ where $k$ is the number of on-time flights, and $\binom{n}{k}$ is the binomial coefficient, or "n choose k."

1. The probability that at least 4 flights are on time:

This requires calculating $P(X \geq 4)$, which means the sum of the probabilities for 4, 5, 6, and 7 flights being on time: $P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)$

2. The probability that at most 5 flights are on time:

This requires calculating $P(X \leq 5)$, which means the sum of the probabilities for 0 through 5 flights being on time: $P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)$

3. The probability that exactly 2 flights are on time:

This is directly calculating $P(X = 2)$ using the binomial probability formula.

Let me compute these probabilities for you.Here are the results:

1. The probability that at least 4 flights are on time is 0.500.
2. The probability that at most 5 flights are on time is 0.938.
3. The probability that exactly 2 flights are on time is 0.164.

Would you like more details on any of these calculations or have any questions?

Here are 5 related questions to explore:

1. What is the probability that all 7 flights are on time?
2. How would the probabilities change if the success rate (on-time probability) was 60% instead of 50%?
3. What is the expected number of on-time flights given the current probability?
4. How would you calculate the standard deviation of the number of on-time flights?
5. What is the probability that exactly 3 flights are on time?

Tip: Binomial distributions are useful when dealing with events that have a fixed number of trials, and each trial has only two possible outcomes.