(c) Consider a batch of 10 randomly selected suspicious transactions. Suppose the specialist wants to know the probability that 2 of the transactions will be blocked. (i) Define the random variable of interest and state how the variable is distributed. (ii) Find the probability that 2 transactions in the batch will be blocked. Show your work.

Let’s dive into the exciting world of probability, shall we? We can define the random variable of interest as \( X \), which represents the number of transactions blocked out of the 10 selected suspicious transactions. Since we are dealing with a fixed number of trials (10 transactions) that either result in success (blocked) or failure (not blocked), and assuming each transaction has the same probability of being blocked, this situation follows a binomial distribution. Specifically, \( X \) is distributed as \( X \sim \text{Binomial}(n = 10, p) \), where \( n \) is the number of trials and \( p \) is the probability of a single transaction being blocked. To calculate the probability that exactly 2 out of these 10 transactions will be blocked, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] In this case, \( k = 2 \), \( n = 10 \), and we need a value for \( p \) (let's say, for the sake of this example, that \( p = 0.3 \)). Plugging in, we find: \[ P(X = 2) = \binom{10}{2} (0.3)^2 (0.7)^{8} \] Calculating \( \binom{10}{2} = 45 \), we have: \[ P(X = 2) = 45 \cdot (0.09) \cdot (0.05764801) \approx 0.2345 \] So, the probability that exactly 2 transactions will be blocked is approximately 0.2345, or 23.45%! Isn't math fun?