To find $P(X \leq 3)$ for a binomial distribution with $n = 14$ and $p = 0.13$, follow these steps using Excel:

1. Open Excel.

2. Use the BINOM.DIST function to calculate the cumulative probability for $X \leq 3$.

   The syntax for the function is:

   =BINOM.DIST(3, 14, 0.13, TRUE)
   

   Where:

   • 3 is the number of successes (for $X \leq 3$).
   • 14 is the number of trials.
   • 0.13 is the probability of success in each trial.
   • TRUE calculates the cumulative probability up to 3.

3. After entering the formula, press Enter. The result will be the probability rounded to four decimal places.

The probability value should be approximately:

$P(X \leq 3) = 0.9361$

Would you like more details on how this formula works or clarification?

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Related Questions:

1. How is the binomial distribution different from other discrete distributions?
2. What is the significance of using cumulative probability in binomial experiments?
3. How would the probability change if $n$ increased while $p$ remained constant?
4. Can we approximate a binomial distribution to a normal distribution, and under what conditions?
5. How can the concept of expected value be applied to binomial distributions?

Tip: When working with probabilities in Excel, always use cumulative functions (like TRUE in BINOM.DIST) to compute values for $X \leq k$ scenarios.