Budget Airlines is a seasonal airline that operates flights from Myrtle Beach, South Carolina, to various cities in the northeast. Recently Budget has been concerned about the number of lost bags. Ann Poston from the Analytics Department was asked to study the issue. She randomly selected a sample of 500 flights and found that a total of twenty bags were lost on the sampled flights. What is the probability that at least 2 bags are lost? Round your answer to four decimal places.

To determine the probability that at least 2 bags are lost on a randomly selected flight, we can use the concept of binomial probability, where:

- \( n = 500 \) (the number of flights),
- \( p = \frac{20}{500} = 0.04 \) (the probability of a bag being lost per flight),
- \( q = 1 - p = 0.96 \) (the probability of no bag being lost per flight).

However, the problem is asking for the probability of at least 2 bags being lost. This is easier to calculate by first finding the probabilities of 0 and 1 bag being lost, and subtracting their sum from 1.

### Step 1: Find the probability that 0 bags are lost.
Using the binomial probability formula:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
Where:
- \( X \) is the number of lost bags,
- \( k \) is the number of successful outcomes (in this case, the number of lost bags),
- \( n \) is the number of trials (in this case, flights),
- \( p \) is the probability of a success (losing a bag).

For 0 lost bags:
\[
P(X = 0) = \binom{500}{0} (0.04)^0 (0.96)^{500} = 1 \cdot 1 \cdot (0.96)^{500}
\]
We'll compute this value numerically.

### Step 2: Find the probability that 1 bag is lost.
Similarly, for 1 lost bag:
\[
P(X = 1) = \binom{500}{1} (0.04)^1 (0.96)^{499} = 500 \cdot 0.04 \cdot (0.96)^{499}
\]
We'll compute this value numerically as well.

### Step 3: Calculate the probability of at least 2 bags lost.
The probability that at least 2 bags are lost is:
\[
P(X \geq 2) = 1 - P(X = 0) - P(X = 1)
\]

Now, let's calculate these probabilities.The probability that at least 2 bags are lost on a randomly selected flight is approximately \( 0.99999997 \), or very close to 1 when rounded to four decimal places.

Would you like more details on this, or have any further questions?

Here are 5 related questions:
1. How would the probability change if the sample size increased to 1000 flights?
2. What is the probability of losing exactly 3 bags on a flight?
3. How do we adjust this calculation for non-binomial distributions?
4. Can this method be used for continuous data? Why or why not?
5. What assumptions are made when applying the binomial distribution here?

**Tip:** Binomial probabilities are great for events with two possible outcomes. If more outcomes exist, consider a multinomial distribution.

Learn how to calculate the probability that at least 2 bags are lost on a randomly selected flight, given a sample of 500 flights with 20 lost bags. This problem involves key probability concepts like the binomial distribution and binomial theorem. Suitable for Grades 11-12.

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