We are given that $X$ is a binomial random variable with $n = 10$ trials and $p = 0.5$ probability of success.

We can calculate the probability for each statement using the binomial probability formula:

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

where $n = 10$ and $p = 0.5$.

Let's evaluate each of the statements one by one:

1. P(X ≤ 2) = 0.0547

The cumulative probability $P(X \leq 2)$ is the sum of the individual probabilities from $P(X = 0)$ to $P(X = 2)$. We can calculate these terms:

• $P(X = 0) = \binom{10}{0} (0.5)^0 (0.5)^{10} = 0.0009765625$
• $P(X = 1) = \binom{10}{1} (0.5)^1 (0.5)^9 = 0.009765625$
• $P(X = 2) = \binom{10}{2} (0.5)^2 (0.5)^8 = 0.0439453125$

Summing these up:

$P(X \leq 2) = 0.0009765625 + 0.009765625 + 0.0439453125 = 0.0546875 \approx 0.0547$

So, this statement is true.

2. P(X = 5) = 0.2461

We calculate $P(X = 5)$ using the binomial formula:

$P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^5 = \binom{10}{5} (0.5)^{10}$

$P(X = 5) = \frac{10!}{5!5!} (0.5)^{10} = 252 \times \frac{1}{1024} = 0.24609375 \approx 0.2461$

So, this statement is true.

3. P(X ≥ 9) = 0.0107

To calculate $P(X \geq 9)$, we need to find the sum of the probabilities for $P(X = 9)$ and $P(X = 10)$:

• $P(X = 9) = \binom{10}{9} (0.5)^9 (0.5)^1 = 10 \times \frac{1}{1024} = 0.009765625$
• $P(X = 10) = \binom{10}{10} (0.5)^{10} = 1 \times \frac{1}{1024} = 0.0009765625$

Summing these:

$P(X \geq 9) = 0.009765625 + 0.0009765625 = 0.0107421875 \approx 0.0107$

So, this statement is true.

4. All of the others

Since we have verified that all of the individual statements are true, this statement is also true.

Conclusion:

The correct answer is All of the others.

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Would you like further clarification on any of these calculations? Here are some follow-up questions for deeper understanding:

1. What is the general binomial probability formula, and how is it derived?
2. How do we calculate cumulative probabilities like $P(X \leq k)$?
3. What is the interpretation of a binomial distribution in real-life examples?
4. How can we use a binomial distribution to model experiments with more trials or different probabilities?
5. What are other methods for calculating binomial probabilities (e.g., using statistical software or tables)?

Tip: When calculating binomial probabilities, it’s helpful to use a calculator or statistical software for quick results, especially when the calculations involve large numbers.