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.

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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.