A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of x successes in the n independent trials of the experiment.

nequals=77​, 

pequals=0.40.4​, 

xless than<44

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Part 1

Upper P left parenthesis Upper X less than 4 right parenthesisP(X<4)equals=enter your response here

​(Round to four decimal places as​ needed.)

To find the probability \( P(X < 4) \) in a binomial probability experiment with the given parameters \( n = 7 \), \( p = 0.4 \), and \( x < 4 \), we can use the binomial cumulative distribution function (CDF).

The formula for the probability mass function (PMF) of a binomial distribution is:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

Where:
- \( X \) is the number of successes,
- \( n \) is the number of trials (7),
- \( p \) is the probability of success on a single trial (0.4),
- \( k \) is the specific number of successes,
- \( \binom{n}{k} \) is the binomial coefficient.

To calculate \( P(X < 4) \), we sum the probabilities for \( X = 0 \), \( X = 1 \), \( X = 2 \), and \( X = 3 \):

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

However, using technology (such as a binomial calculator or software like Excel, Python, or a statistical calculator) to find the cumulative probability \( P(X < 4) \) is much faster.

Using a binomial distribution calculator:

- Set \( n = 7 \),
- Set \( p = 0.4 \),
- Find \( P(X < 4) \).

This gives the cumulative probability.

Let me compute this for you.The probability \( P(X < 4) \) for the binomial distribution with \( n = 7 \), \( p = 0.4 \) is approximately \( 0.7102 \) (rounded to four decimal places).

Would you like further details or explanations?

Here are five related questions you can explore:
1. How would the probability change if \( p \) were 0.6 instead of 0.4?
2. What is \( P(X \geq 4) \) in this same binomial experiment?
3. How is the binomial distribution connected to the normal distribution?
4. What happens to \( P(X < 4) \) if \( n \) is increased to 10?
5. Can you explain the difference between the binomial and geometric distributions?

**Tip:** The binomial distribution is useful for discrete events with two possible outcomes, making it ideal for modeling success/failure experiments.

A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of x successes in the n independent trials of the experiment. n = 7, p = 0.4, x < 4. P(X < 4) (Round to four decimal places as needed.)

Explore how to calculate the binomial probability P(X < 4) for a distribution with n = 7 trials and p = 0.4 probability of success. This detailed guide covers the binomial probability formula and the use of technology to find cumulative probabilities. Ideal for Grades 10-12.

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