What is the probability of obtaining exactly 11 success in 33 trials when the probability of success is 0.80.8.

Round your answer to four decimal places.
 

To calculate the probability of obtaining exactly 11 successes in 33 trials when the probability of success is 0.8, we can use the binomial probability formula:

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

where:

- \( n = 33 \) (number of trials)
- \( a = 11 \) (number of successes)
- \( p = 0.8 \) (probability of success on a single trial)

Let's compute the probability:

1. **Calculate the binomial coefficient \(\binom{33}{11}\):**

\[
\binom{33}{11} = \frac{33!}{11!(33-11)!}
\]

2. **Compute \( p^a \) and \( (1-p)^{n-a} \):**

\[
p^{11} = (0.8)^{11}
\]

\[
(1-p)^{33-11} = (0.2)^{22}
\]

3. **Multiply all components together:**

\[
P(X = 11) = \binom{33}{11} \times (0.8)^{11} \times (0.2)^{22}
\]

I’ll calculate this step-by-step.

\[
\binom{33}{11} = \frac{33!}{11! \times 22!} \approx 2100340
\]

\[
(0.8)^{11} \approx 0.0859
\]

\[
(0.2)^{22} \approx 4.194 \times 10^{-16}
\]

Now, multiply all components:

\[
P(X = 11) = 2100340 \times 0.0859 \times 4.194 \times 10^{-16}
\]

\[
P(X = 11) \approx 7.5731 \times 10^{-14}
\]

So, the binomial probability is approximately **\(7.5731 \times 10^{-14}\)**.

Would you like more details or have any questions?

Here are 5 related questions you might be interested in:

1. What is the probability of obtaining 10 or fewer successes in 33 trials?
2. How does the probability change if the number of trials is increased to 50?
3. What is the expected number of successes in this binomial distribution?
4. How would you compute the probability for a different number of successes, say 15?
5. How can you use the normal approximation to calculate this binomial probability?

**Tip:** For large combinations, using software or a calculator with a binomial function can help avoid calculation errors and save time.

Learn how to calculate the probability of exactly 11 successes in 33 trials with a success probability of 0.8 using the binomial probability formula. Step-by-step solution provided.

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