Final answer:
In the field of Mathematics, specifically probability and binomial distribution, to determine the likelihood that a binomially distributed random variable X exceeds 9 with n = 25 and p = 0.25, one could utilize the normal approximation and calculate P(X > 9.5) using the appropriate z-scores after applying a continuity correction.
Step-by-step explanation:
The subject of this question is Mathematics, specifically in the area of probability dealing with the binomial distribution. Given are the parameters for a binomial distribution where the number of trials (n) is 25 and the probability of success (p) is 0.25. To calculate the probability that the random variable X is greater than 9, we need to find P(X > 9). To approach this, we can use the cumulative distribution function (cdf) for the binomial distribution and subtract from 1 to find the complement. In other words, P(X > 9) = 1 - P(X ≤ 9). However, our question fits the rule stating that the binomial distribution can be approximated by a normal distribution if the products np and nq are greater than five. In our case, np = 25(0.25) = 6.25 and nq = 25(0.75) = 18.75 both satisfy this rule. Therefore, we can use the normal approximation to the binomial distribution to find the probability. We would convert our binomial problem to a normal distribution problem and use z-scores to find the probability P(X > 9.5) since we add 0.5 for continuity correction when dealing with discrete distributions like the binomial.