To solve for $P(X \leq k)$ for a binomial distribution, we use the binomial cumulative probability formula:

$P(X \leq k) = \sum_{x=0}^{k} \binom{n}{x} p^x (1-p)^{n-x}$

where:

• $n$ is the number of trials
• $p$ is the probability of success
• $k$ is the number of successes we are interested in
• $\binom{n}{x}$ is the binomial coefficient

Let’s work through each part step-by-step.

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(a) $n = 20, p = 0.35, k = 2$

We need to calculate $P(X \leq 2)$ for a binomial distribution with $n = 20$, $p = 0.35$, and $k = 2$.

Using a calculator or statistical software, we can find:

$P(X \leq 2) \approx 0.0401$

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(b) $n = 15, p = 0.8, k = 8$

We need to calculate $P(X \leq 8)$ for a binomial distribution with $n = 15$, $p = 0.8$, and $k = 8$.

Using the same approach:

$P(X \leq 8) \approx 0.0005$

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(c) $n = 10, p = 0.7, k = 9$

We need to calculate $P(X \leq 9)$ for a binomial distribution with $n = 10$, $p = 0.7$, and $k = 9$.

From a binomial cumulative probability calculator:

$P(X \leq 9) \approx 0.9719$

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Summary of answers:

• (a) $P(X \leq 2) \approx 0.040$
• (b) $P(X \leq 8) \approx 0.0005$
• (c) $P(X \leq 9) \approx 0.972$

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Would you like to dive deeper into how to compute these by hand or need any further clarification?

Related questions:

1. How do I compute binomial probabilities manually without using software?
2. What is the difference between a binomial and a normal distribution in terms of cumulative probability?
3. How can I use the normal approximation to the binomial distribution?
4. What is the binomial coefficient $\binom{n}{x}$, and how is it calculated?
5. How do I find $P(X = k)$ for a binomial distribution?

Tip: When working with binomial probabilities, using a calculator or statistical software like Excel or R can significantly speed up the process!