¿ué operación se utiliza para caicular cierta probabilidad de más de un evento cuando se avanza hacia abajo en el diagrama de árbol? a suma multiplicación c. división d. resta

4 Permutations and Probabilities: 20 pts, Part A: 10 pts. You are tasked with organizing a group of 6 people: Allice, Bob, Charlie, David, Eva, and Frank. How many different ways can you arrange these 6 people in a line? Show your work to receive full credit. Part B: 10 pts. If the arrangement from Part A must have Alice and Bob standing next to each other, what is the probability that Alice and Bob will be next to each other in a random arrangement? Show your work to receive full credit.

After a particular surgery, patients are placed in an intensive care unit (ICU) until their condition stabilizes. Then they are transferred to a special care ward (SCW), where they remain until they are released from the hospital. In a particular metropolitan area, a study of hospital records produced the following data: each day \( 2 \% \) of the patients in the ICU died, \( 49 \% \) were transferred to the SCW, and the remainder stayed in the ICU. Furthermore, each day \( 3 \% \) of the patients in the SCW developed complications and were returned to the ICU, \( 1 \% \) died while in the SCW, \( 22 \% \) were released from the hospital, and the remainder stayed in the SCW. A) In the long run, what percentage of the patients in the ICU are released from the hospital? \( \square \% \) (Round to two decimal places as needed.)

3. En un míni mercado se tienen 2 cajeros que siempre están trabajando. La llegada de clientes al míni mercado es de 13 por hora y cada cajero atiende un promedio de los clientes cada 15 minutos. a. Halle Probabilidad de que el sistema esté vacío \( \left(P_{0}\right) \) b. Cuál es el promedio de clientes en el sistema. \( \left(L_{s}\right) \) c. Cuál es el promedio de clientes en la cola. \( \left(L_{q}\right) \) d. Cuál es el promedio del sistema. \( \left(W_{s}\right) \) e. Tiempo promedio en cola. \( \left(W_{4}\right) \)

3. Combinations and Probabilities: 20 pts. You are tasked with creating a code using the following three conditions: 1. The code must consist of exactly 4 characters. 2. Each character can be one of the following: A, B, C, or D. 3. The code can repeat characters, but the first character must be different from the last character. How many different codes can you create based on these condi- tions? You MUST show all of your steps to receive full credit.

logía, Química, Física y Lenguaje. Si se toma uno de ellos, ¿cuá a probabilidad de que este sea de matemática o de física? \( 1 / 5 \) b) \( 3 / 5 \) c) \( 2 / 5 \mathrm{~d} / 4 / 5 \)