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.