Let X represent a binomial random variable with n = 180 and p = 0.23. Find the following probabilities. (Do not round intermediate calculations. Round your final answers to 4 decimal places.)

To find probabilities for a binomial random variable \( X \), where \( X \sim \text{Binomial}(n = 180, p = 0.23) \), we can use the binomial probability formula:

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

Where:
- \( n = 180 \) is the number of trials,
- \( p = 0.23 \) is the probability of success on a single trial,
- \( k \) is the number of successes,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

Please specify the exact probabilities you need (e.g., \( P(X = k) \), \( P(X \leq k) \), \( P(X \geq k) \), etc.), and I'll calculate those for you!

Let me know if you want to go ahead with a specific calculation or need help with particular values.

Learn how to calculate probabilities for a binomial random variable X with n = 180 and p = 0.23. This problem uses the binomial probability formula and covers key probability concepts. Suitable for advanced high school students and undergraduates.

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