Given a binomial distribution in which the probability of success is 0.77 and the number of trials is 14, what is the probability for each of the following:

*(Round your answers to 3 decimal places)*

a) Getting exactly 12 successes?

b) Getting more than 12 successes?

c) Getting less than or equal to 12 successes?

To solve these questions, we'll use the binomial probability formula:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

where:
- \(n\) is the number of trials,
- \(k\) is the number of successes,
- \(p\) is the probability of success on a single trial,
- \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

Given:
- \(n = 14\)
- \(p = 0.77\)
- \(q = 1 - p = 0.23\)

### a) Getting exactly 12 successes

Here, \(k = 12\).

\[
P(X = 12) = \binom{14}{12} \cdot (0.77)^{12} \cdot (0.23)^{2}
\]

### b) Getting more than 12 successes

This is the sum of the probabilities of getting 13 or 14 successes:

\[
P(X > 12) = P(X = 13) + P(X = 14)
\]

### c) Getting less than or equal to 12 successes

This is the sum of the probabilities of getting 0 to 12 successes:

\[
P(X \leq 12) = \sum_{k=0}^{12} P(X = k)
\]

Let's calculate these values.Here are the probabilities rounded to three decimal places:

a) The probability of getting exactly 12 successes is \( P(X = 12) = 0.209 \).

b) The probability of getting more than 12 successes is \( P(X > 12) = 0.133 \).

c) The probability of getting less than or equal to 12 successes is \( P(X \leq 12) = 0.867 \).

Would you like any further details or have any questions?

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**Related Questions:**

1. What is the probability of getting exactly 13 successes?
2. How does the probability change if the number of trials increases to 20?
3. What is the expected number of successes in 14 trials?
4. How do you calculate the standard deviation for this binomial distribution?
5. What is the cumulative probability of getting fewer than 10 successes?
6. How would the probability of exactly 12 successes change if the success probability was 0.8?
7. What is the probability of getting at least 10 successes?
8. How would you interpret the probability results in a real-world scenario?

**Tip:** Understanding the symmetry in binomial distributions can often simplify probability calculations, especially for large \(n\) or when \(p\) is close to 0.5.

Calculate the probabilities for a binomial distribution with 14 trials and a success probability of 0.77. Learn how to find the probability of exactly 12 successes, more than 12 successes, and less than or equal to 12 successes using the binomial probability formula. Round answers to three decimal places.

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