It has been reported that 40% of college students graduate in 4 years.
Find the probability that in a random sample of 30 students at most 18 students will graduate in 4 years and at most 18 students will fail to graduate in 4 years.

To find the probability that in a random sample of 30 students at most 18 students will graduate in 4 years and at most 18 students will fail to graduate in 4 years, we need to approach this problem using binomial probability.

Let’s break down the problem:

1. The probability \( p \) that a student graduates in 4 years is 0.4.
2. The sample size \( n \) is 30 students.
3. We need to find the probability of at most 18 students graduating in 4 years.
4. We also need to find the probability of at most 18 students not graduating in 4 years.

### Step 1: Probability that at most 18 students graduate in 4 years

The number of students graduating in 4 years, \( X \), follows a binomial distribution \( X \sim \text{Binomial}(n = 30, p = 0.4) \).

The probability we need to find is \( P(X \leq 18) \).

Using the binomial cumulative distribution function (CDF):

\[ P(X \leq 18) = \sum_{k=0}^{18} \binom{30}{k} (0.4)^k (0.6)^{30-k} \]

### Step 2: Probability that at most 18 students fail to graduate in 4 years

The number of students not graduating in 4 years, \( Y \), follows a binomial distribution \( Y \sim \text{Binomial}(n = 30, p = 0.6) \).

The probability we need to find is \( P(Y \leq 18) \).

Using the binomial cumulative distribution function (CDF):

\[ P(Y \leq 18) = \sum_{k=0}^{18} \binom{30}{k} (0.6)^k (0.4)^{30-k} \]

### Calculations

We'll calculate these probabilities using a binomial calculator or statistical software. However, the calculations for these can be done manually or using a software tool like Python. 

#### For \( P(X \leq 18) \):

\[
P(X \leq 18) = 1 - P(X > 18)
\]

Using binomial tables or a calculator, we can find \( P(X > 18) \).

\[
P(X \leq 18) \approx 1 - P(X > 18)
\]

#### For \( P(Y \leq 18) \):

\[
P(Y \leq 18) = 1 - P(Y > 18)
\]

Using binomial tables or a calculator, we can find \( P(Y > 18) \).

\[
P(Y \leq 18) \approx 1 - P(Y > 18)
\]

### Using Python to Calculate Probabilities

To get accurate results, we'll use Python with the SciPy library.

```python
from scipy.stats import binom

# Parameters
n = 30
p_graduate = 0.4
p_fail = 0.6

# Calculate probabilities
P_X_leq_18 = binom.cdf(18, n, p_graduate)
P_Y_leq_18 = binom.cdf(18, n, p_fail)

P_X_leq_18, P_Y_leq_18
```

The output will give us the probabilities:

1. \( P(X \leq 18) \)
2. \( P(Y \leq 18) \)

Both probabilities will be approximately the same due to symmetry.

### Conclusion

The probabilities will give us the required likelihoods for both scenarios.

Would you like details or have any questions?

Here are some related questions you might find interesting:

1. What is the probability that exactly 12 students will graduate in 4 years?
2. How would the probabilities change if the sample size were 50 students instead of 30?
3. What is the expected number of students who will graduate in 4 years from a sample of 30?
4. How would you calculate the probability that more than 20 students will graduate in 4 years?
5. Can you find the probability distribution for the number of students graduating in 4 years?
6. What is the variance in the number of students graduating in 4 years from a sample of 30?
7. How would the calculations change if the graduation rate was 50% instead of 40%?
8. What is the probability that at least 10 students will fail to graduate in 4 years from a sample of 30?

**Tip:** Binomial distributions are useful for modeling scenarios with a fixed number of independent trials and binary outcomes, such as success/failure or graduate/not graduate.

Learn how to calculate the probability that in a random sample of 30 college students, at most 18 will graduate in 4 years and at most 18 will fail to graduate in 4 years. Understand binomial distribution and probability calculations.

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