There are 8 acts in a talent show. An acrobat, a dancer, a guitarist, a juggler, a magician, a pianist, a violinist, and a whistler. A talent show host randomly schedules the 8 acts. Compute the probability, of each of the following events. Event A : The first four acts are the guitarist, the pianist, the juggler, and the violinist, in any order. Event B : The juggler is first, the guitarist is second, the violinist is third, and the pianist is fourth. Write your answers as fractions in simplest form. \( P(A)= \)

To compute the probability of Event A and Event B, we will start with Event A. 1. **Event A**: The first four acts are the guitarist, the pianist, the juggler, and the violinist, in any order. First, we calculate the total number of ways to arrange all 8 acts. The total arrangements (permutations) of 8 acts is given by \(8!\): \[ 8! = 40320 \] Next, we find the number of favorable outcomes for Event A. The first four acts can be arranged in any order. The number of arrangements of the four acts is \(4!\): \[ 4! = 24 \] The remaining four acts (the acrobat, dancer, magician, and whistler) can also be arranged in any order, which gives us another \(4!\): \[ 4! = 24 \] So, the total number of favorable arrangements for Event A is: \[ 4! \times 4! = 24 \times 24 = 576 \] Now, we can calculate the probability of Event A: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{576}{40320} \] Simplifying this fraction, we notice that both numbers can be divided by 576: \[ P(A) = \frac{1}{70} \] Now for **Event B**: The juggler is first, the guitarist is second, the violinist is third, and the pianist is fourth. In this case, since the first four positions are fixed to specific acts, there are no additional arrangements for those positions; they are predetermined. For the remaining four acts (acrobat, dancer, magician, and whistler), they can be arranged in: \[ 4! = 24 \] Thus, the total number of favorable outcomes for Event B is simply: \[ 24 \] Now we calculate the probability of Event B: \[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{24}{40320} \] Simplifying this fraction, we find that both numbers can be divided by 24: \[ P(B) = \frac{1}{1680} \] Finally, the probabilities are: \[ P(A) = \frac{1}{70} \quad \text{and} \quad P(B) = \frac{1}{1680} \]