Compute the binomial probability:
n
=
6
,
a
=
5
,
p
=
0.75
)

Round your answer to four decimal places.

To compute the binomial probability, we can use the binomial probability formula:

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

where:

- \( n = 6 \) (number of trials)
- \( a = 5 \) (number of successes)
- \( p = 0.75 \) (probability of success on a single trial)

Let's calculate the binomial probability:

\[
P(X = 5) = \binom{6}{5} (0.75)^5 (1 - 0.75)^{6-5}
\]

First, calculate the binomial coefficient \(\binom{6}{5}\):

\[
\binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6 \times 5!}{5! \times 1!} = 6
\]

Next, calculate the individual components:

\[
(0.75)^5 = 0.2373
\]

\[
(1 - 0.75)^{6-5} = 0.25^1 = 0.25
\]

Now, multiply all the components together:

\[
P(X = 5) = 6 \times 0.2373 \times 0.25 = 0.35595
\]

Finally, rounding to four decimal places:

\[
P(X = 5) \approx 0.3560
\]

So, the binomial probability is approximately **0.3560**.

Would you like more details or have any questions?

Here are 5 related questions you might be interested in:

1. How do you compute the cumulative binomial probability for \( P(X \leq 5) \)?
2. What is the expected value of a binomial distribution with \( n = 6 \) and \( p = 0.75 \)?
3. How does changing the probability \( p \) affect the binomial probability?
4. Can you compute the probability for \( P(X = 6) \) given the same parameters?
5. How do you derive the binomial coefficient formula?

**Tip:** Always double-check your binomial coefficient calculation, as it is crucial for accurate binomial probability results.

Learn how to compute the binomial probability with n = 6, a = 5, and p = 0.75 using the binomial probability formula. Understand the calculation steps involving binomial coefficients, probability of success, and number of trials. Suitable for advanced high school students.

Math Problem Statement

Compute the binomial probability: n

6 , a

5 , p

Solution

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