Consider a binomial experiment with N=20 and P=0.70 .

**a.** Compute f(12)  (to 4 decimals).
 
**b.** Compute f(16) (to 4 decimals).

In a binomial experiment, the probability of observing exactly \( k \) successes in \( N \) trials, where each trial has a probability \( P \) of success, is given by the formula:

\[
f(k) = \binom{N}{k} P^k (1 - P)^{N - k}
\]

where:
- \( N = 20 \),
- \( P = 0.70 \),
- \( k = 12 \) (for part a), and \( k = 16 \) (for part b).

The binomial coefficient \(\binom{N}{k}\) is calculated as:

\[
\binom{N}{k} = \frac{N!}{k!(N - k)!}
\]

Let's calculate each part.

### **Part (a): Compute \( f(12) \)**
For \( k = 12 \):
\[
f(12) = \binom{20}{12} (0.70)^{12} (0.30)^{20 - 12}
\]

### **Part (b): Compute \( f(16) \)**
For \( k = 16 \):
\[
f(16) = \binom{20}{16} (0.70)^{16} (0.30)^{20 - 16}
\]

I will calculate these values.The results are as follows:

**a.** \( f(12) = 0.1144 \) (to 4 decimal places)

**b.** \( f(16) = 0.1304 \) (to 4 decimal places)

Would you like more details or have any questions about this? 

Here are some related questions to consider:

1. How does changing \( N \) or \( P \) affect the probability outcomes?
2. What is the cumulative probability of observing at most 12 successes in this experiment?
3. How would we compute the probability of getting between 10 and 15 successes?
4. What is the expected value (mean) and variance of this binomial distribution?
5. How does this probability calculation change if we increase the probability \( P \) to 0.80?

**Tip:** In binomial experiments, increasing \( N \) or \( P \) can significantly change the distribution's shape, often concentrating probabilities around higher values of \( k \) when \( P \) is high.

Consider a binomial experiment with N=20 and P=0.70. 

**a.** Compute f(12) (to 4 decimals).

**b.** Compute f(16) (to 4 decimals).

Learn how to calculate binomial probabilities for a binomial experiment with N=20 and probability of success P=0.70. Find the exact probabilities for 12 and 16 successes, presented with step-by-step solutions suitable for high school students.

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