In this problem, we are dealing with a binomial probability experiment, where:

• $n = 9$ (the number of trials),
• $p = 0.8$ (the probability of success in each trial),
• and we are asked to find the probability of having $x \leq 3$ (3 or fewer successes).

The probability mass function for a binomial distribution is:

$P(X = x) = \binom{n}{x} \cdot p^x \cdot (1 - p)^{n - x}$

where:

• $\binom{n}{x}$ is the binomial coefficient (the number of ways to choose $x$ successes from $n$ trials),
• $p^x$ is the probability of $x$ successes,
• $(1 - p)^{n - x}$ is the probability of $n - x$ failures.

To solve for $P(X \leq 3)$, we need to calculate:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Let's compute these probabilities step by step.

I'll calculate that now.The probability of having $x \leq 3$ successes in 9 independent trials, with a success probability of 0.8, is approximately 0.0031 (rounded to four decimal places).

Would you like further details on this, or do you have any other questions?

Here are five related questions for deeper understanding:

1. How would the probability change if $p = 0.6$ instead of 0.8?
2. What is the probability of exactly 4 successes in the same experiment?
3. How do you calculate the expected number of successes in a binomial experiment?
4. How would you compute the cumulative probability for $x \leq 5$?
5. What is the standard deviation of this binomial distribution?

Tip: For binomial distributions, if the number of trials $n$ increases, the distribution tends to look more like a normal distribution (Central Limit Theorem).