Certainly! Let's break down the calculations for each part.

### Part 1: Probability of exactly 6 successes

The formula for the binomial probability of exactly \( k \) successes is:

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

where:
- \( n = 16 \) is the number of trials,
- \( k = 6 \) is the number of successes we are interested in,
- \( p = 0.45 \) is the probability of success on each trial.

First, we calculate the binomial coefficient \( \binom{16}{6} \):

\[
\binom{16}{6} = \frac{16!}{6!(16-6)!} = \frac{16!}{6! \times 10!}
\]

Next, we calculate the probability:

\[
P(X = 6) = \binom{16}{6} \times (0.45)^6 \times (0.55)^{10}
\]

Breaking it down:

- \( \binom{16}{6} = 8008 \)
- \( (0.45)^6 \approx 0.00895 \)
- \( (0.55)^{10} \approx 0.00479 \)

So:

\[
P(X = 6) = 8008 \times 0.00895 \times 0.00479 \approx 0.168
\]

### Part 2: Probability of at least 10 successes

To find \( P(X \geq 10) \), we sum the probabilities for \( X = 10 \) through \( X = 16 \):

\[
P(X \geq 10) = P(X = 10) + P(X = 11) + \dots + P(X = 16)
\]

Alternatively, this can be calculated using the cumulative distribution function (CDF) of the binomial distribution:

\[
P(X \geq 10) = 1 - P(X \leq 9)
\]

Using the CDF:

- Calculate \( P(X \leq 9) \), which sums the probabilities from \( X = 0 \) to \( X = 9 \).
- Subtract this from 1 to find \( P(X \geq 10) \).

This gives us:

\[
P(X \geq 10) \approx 1 - 0.876 \approx 0.124
\]

### Part 3: Probability of less than 11 successes

To find \( P(X < 11) \), we sum the probabilities for \( X = 0 \) through \( X = 10 \):

\[
P(X < 11) = P(X = 0) + P(X = 1) + \dots + P(X = 10)
\]

This is the same as the CDF at \( X = 10 \):

\[
P(X < 11) = P(X \leq 10)
\]

Using the CDF:

\[
P(X < 11) = 0.951
\]

### Final Results:

- **(a)** Probability of exactly 6: \( P(X = 6) \approx 0.168 \)
- **(b)** Probability of at least 10: \( P(X \geq 10) \approx 0.124 \)
- **(c)** Probability of less than 11: \( P(X < 11) \approx 0.951 \)

These calculations match the earlier values provided. Would you like further clarifications or to explore related questions?

Explore how to calculate probabilities in a binomial distribution problem involving 16 trials with a success probability of 0.45. Learn the formulas for exact probabilities, cumulative probabilities, and more. Detailed solution provided.

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