To solve the problem, we first need to determine the probability of spinning an "A" on a single spin of the spinner.

Step 1: Calculate the Probability of Spinning "A" on One Spin

The spinner has 8 equal parts with the following distribution: B, A, B, C, A, B, C, C. Out of these 8 parts, "A" appears 2 times.

So, the probability $P(A)$ of landing on "A" in a single spin is:

$P(A) = \frac{\text{Number of A's on the spinner}}{\text{Total number of sections on the spinner}} = \frac{2}{8} = \frac{1}{4} = 0.25$

Step 2: Apply the Binomial Probability Formula

We want to find the probability of getting "A" exactly 16 times out of 38 spins. This is a binomial probability problem where:

• $n = 38$ (number of trials)
• $k = 16$ (number of successes, i.e., number of times "A" is spun)
• $p = 0.25$ (probability of success on a single trial)

The binomial probability formula is:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$
• $p^k$ is the probability of getting exactly $k$ successes
• $(1-p)^{n-k}$ is the probability of the remaining $n-k$ trials being failures

Step 3: Calculate the Probability

$P(X = 16) = \binom{38}{16} (0.25)^{16} (0.75)^{22}$

1. Calculate the binomial coefficient $\binom{38}{16}$:

$\binom{38}{16} = \frac{38!}{16!(38-16)!} = \frac{38!}{16! \times 22!}$

2. Calculate $(0.25)^{16}$ and $(0.75)^{22}$.

3. Multiply these together to get the final probability.

Step 4: Compute the Final Result

Let me calculate it for you.The probability of spinning "A" exactly 16 times out of 38 spins is approximately 0.0092 or 0.92%.

This means that under these conditions, there is less than a 1% chance of landing on "A" exactly 16 times out of 38 spins.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the probability of spinning "A" at least 16 times?
2. How does the probability change if the spinner is spun 50 times instead of 38?
3. What is the expected number of times to spin "A" in 38 spins?
4. What is the variance and standard deviation of spinning "A" in 38 spins?
5. How does the probability change if the spinner is fair but with different sections?
6. What if we want the probability of spinning "A" fewer than 10 times?
7. How does changing the number of sections affect the probability of a specific outcome?
8. What is the probability of not spinning "A" at all in 38 spins?

Tip: In probability problems, identifying the type of distribution (binomial, normal, etc.) is crucial to using the correct formula.