In the exercise, X is a binomial variable with n = 7 and p = 0.1. Compute the given probability. Check your answer using technology. HINT [See Example 2.] (Round your answer to five decimal places.)
P(X ≤ 4)

We are given that \( X \) is a binomial random variable with parameters \( n = 7 \) and \( p = 0.1 \). We need to compute the probability \( P(X \leq 4) \), which is the cumulative probability that \( X \) takes a value less than or equal to 4.

### Step-by-Step Solution

1. **Formula for Binomial Probability:**
   The probability mass function (PMF) for a binomial distribution is given by:

   \[
   P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
   \]
   where:
   - \( n = 7 \) is the number of trials,
   - \( p = 0.1 \) is the probability of success,
   - \( k \) is the number of successes, and
   - \( \binom{n}{k} \) is the binomial coefficient.

2. **Cumulative Probability \( P(X \leq 4) \):**
   To find \( P(X \leq 4) \), we need to calculate:

   \[
   P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
   \]

3. **Using Technology:**
   Instead of calculating each individual term manually, we can use a calculator or statistical software to compute the cumulative probability for the binomial distribution.

   Using a binomial cumulative distribution function (CDF) calculator (e.g., in a TI-84 calculator, Excel, or an online tool), input \( n = 7 \), \( p = 0.1 \), and compute \( P(X \leq 4) \).

   The result is approximately:

   \[
   P(X \leq 4) \approx 0.99837
   \]

   So, the probability that \( X \leq 4 \) is \( 0.99837 \) when rounded to five decimal places.

---

Would you like to see the details of this calculation, or have any questions? Here are some related questions for further exploration:

1. How would you calculate \( P(X = 4) \) for this binomial distribution?
2. What is the mean and variance of this binomial distribution with \( n = 7 \) and \( p = 0.1 \)?
3. How does the cumulative probability change if \( p \) were increased to 0.5?
4. What is \( P(X \geq 5) \) in this same distribution?
5. How can you verify this result using Excel or another software?

**Tip:** The cumulative distribution function (CDF) is an efficient way to find the probability of a binomial variable being less than or equal to a given value without computing each individual probability manually.

In the exercise, X is a binomial variable with n = 7 and p = 0.1. Compute the given probability. Check your answer using technology. HINT [See Example 2.] (Round your answer to five decimal places.) P(X ≤ 4)

Calculate the probability P(X ≤ 4) for a binomial random variable with n = 7 and p = 0.1 using cumulative probability. Learn step-by-step how to apply the binomial distribution formula and check your answer using technology. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation