To solve this problem, we need to find the probability that 15 or more flights from a sample of 18 arrive on time, given that the probability of a flight arriving on time is 74% (or 0.74).

Since the problem states that the number of flights arriving on time follows a binomial distribution, we will use the binomial probability formula:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$

Where:

• $n = 18$ (the number of trials or flights),
• $p = 0.74$ (the probability of a flight arriving on time),
• $k$ is the number of successful outcomes (flights arriving on time), and
• $\binom{n}{k}$ is the binomial coefficient, which can be computed as:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

We are looking for the probability that 15 or more flights arrive on time. This means we need to compute:

$P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18)$

We can either compute each binomial probability separately and sum them or use a binomial cumulative distribution function (CDF). It’s often easier to use a calculator or statistical software to compute this efficiently.

Let’s first compute the cumulative probability for $X \leq 14$ and subtract it from 1 to get $P(X \geq 15)$.

Step 1: Find $P(X \leq 14)$

We can calculate $P(X \leq 14)$ using a binomial cumulative distribution function:

$P(X \leq 14) = \text{binom.cdf}(14, 18, 0.74)$

Step 2: Find $P(X \geq 15)$

Now, we subtract the result from 1:

$P(X \geq 15) = 1 - P(X \leq 14)$

I’ll do the calculation for you. scipy.stats import binom

Given parameters

n = 18 # number of trials (flights) p = 0.74 # probability of success (on-time arrival)

Calculate cumulative probability for X <= 14

P_X_leq_14 = binom.cdf(14, n, p)

Calculate P(X >= 15) = 1 - P(X <= 14)

P_X_geq_15 = 1 - P_X_leq_14 round(P_X_geq_15, 3)The probability that 15 or more flights from the sample of 18 will arrive on time is approximately 0.273.

Would you like further clarification or additional questions on this topic?

Here are some related questions you may find interesting:

1. How would the probability change if the on-time arrival rate were 80% instead of 74%?
2. How can we calculate the expected number of flights arriving on time from this sample?
3. What is the standard deviation of the number of on-time flights in this case?
4. If we wanted to calculate the probability of exactly 12 flights arriving on time, how would we do that?
5. How does the binomial distribution compare to the normal distribution for large $n$?

Tip: When working with binomial distributions, using a cumulative distribution function (CDF) is often much quicker than calculating each individual probability, especially for large values of $n$.