A binomial probability experiment is conducted with the given parameters. Use StatCrunch to find the probability of x successes in the n independent trials of the experiment.
nequals8​, pequals0.25​, xless than4
Question content area bottom
Part 1
Upper P left parenthesis Upper X less than 4 right parenthesisequals

enter your response here
​(Round to four decimal places as​ needed.)

To find the probability \( P(X < 4) \) for a binomial distribution where \( n = 8 \), \( p = 0.25 \), and \( x \) is less than 4, we need to calculate the cumulative probability for \( x = 0, 1, 2, 3 \).

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 \( \frac{n!}{k!(n-k)!} \),
- \( p = 0.25 \) is the probability of success,
- \( 1 - p = 0.75 \) is the probability of failure,
- \( n = 8 \) is the number of trials, and
- \( k \) is the number of successes (for each value of \( k = 0, 1, 2, 3 \)).

Alternatively, using a calculator or a statistical tool like StatCrunch, you can compute this directly by summing the binomial probabilities for \( X = 0, 1, 2, 3 \).

### Using StatCrunch:
1. Open StatCrunch.
2. Navigate to **Stat > Calculators > Binomial**.
3. Enter the values: \( n = 8 \), \( p = 0.25 \).
4. Select **P(X < 4)**.

### Answer:
The probability \( P(X < 4) \) is approximately **0.8860** (rounded to four decimal places).

Would you like more details or further explanation?

Here are some related questions:
1. How do you interpret binomial probabilities in real-life scenarios?
2. What is the difference between cumulative probability and regular probability?
3. How would the probability change if the number of trials increased to 10?
4. What is the impact of increasing the probability of success \( p \) in binomial experiments?
5. How do you use factorials to calculate binomial coefficients manually?

**Tip:** In a binomial experiment, the probability of success and failure adds up to 1 (i.e., \( p + q = 1 \)).

A binomial probability experiment is conducted with the given parameters. Use StatCrunch to find the probability of x successes in the n independent trials of the experiment. n = 8, p = 0.25, x < 4.

Learn how to solve a binomial probability problem with n = 8 and p = 0.25, calculating the cumulative probability of x < 4. This solution covers the binomial probability formula, cumulative probability, and how to use StatCrunch for accurate results. Suitable for students in 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