$$  $$What is the cumulative probability that at most 4 passengers show up?

To find the cumulative probability that at most 4 passengers show up, we sum the individual probabilities for $ X = 0, 1, 2, 3, $ and $ 4 $. We already computed these individual probabilities in the previous solution:

1. **$ P(X = 0) $**:
   \[
   P(X = 0) = 0.000128
   \]

2. **$ P(X = 1) $**:
   \[
   P(X = 1) = 0.003584
   \]

3. **$ P(X = 2) $**:
   \[
   P(X = 2) = 0.043008
   \]

4. **$ P(X = 3) $**:
   \[
   P(X = 3) = 0.1835008
   \]

5. **$ P(X = 4) $**:
   \[
   P(X = 4) = 0.114688
   \]

### Step 1: Sum the probabilities.

\[
P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
\]
\[
P(X \leq 4) = 0.000128 + 0.003584 + 0.043008 + 0.1835008 + 0.114688 = 0.3449088
\]

### Step 2: Round the result to two decimal places.

\[
P(X \leq 4) \approx 0.34
\]

So, the cumulative probability that at most 4 passengers will show up is **0.34**.

Would you like any further details or have any questions?

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Here are some related questions:
1. How would you calculate $ P(X \geq 5) $ for this scenario?
2. What is the probability that exactly 6 passengers show up?
3. How does the probability distribution change as $ n $ increases?
4. What is the significance of the cumulative distribution function (CDF) in probability?
5. How can binomial distribution be applied to real-world scenarios?
6. What tools or software can be used to compute binomial probabilities efficiently?
7. How would the probability be affected if the show-up rate was 70% instead of 80%?
8. Can this problem be solved using the normal approximation to the binomial distribution?

**Tip:** When dealing with cumulative probabilities, make sure to sum the probabilities for each relevant outcome (in this case, from $ X = 0 $ to $ X = 4 $) to ensure accuracy.

Learn how to calculate the cumulative probability that at most 4 passengers show up in a scenario with an 80% show-up rate. Detailed step-by-step solution using the binomial distribution formula. Suitable for undergraduate students and those interested in probability theory.

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