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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.

United Airlines' flights from Boston to Seattle are on time 50% of the time. Suppose seven flights are randomly selected, and the number of on-time flights is recorded. Calculate the probabilities for different cases.

This problem explores the binomial probability distribution for United Airlines flights from Boston to Seattle. It covers calculating the probabilities of at least four flights, at most five flights, and exactly two flights being on time. Suitable for Grades 10-12.

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