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.