En un juego de apuestas entre los estudiantes de Ingenieria en Minas de la UTH y el profesor de Probabilidad y Estadistica, un estudiante lanza una moneda diez veces. En este juego se tienen en cuenta las siguientes apuestas: a) Si obtiene 5 caras o más, el estudiante obtiene 2 puntos en su examen de la unidad. b) Si obtiene menos de tres caras, no gana ningún punto. c) Si obtiene más de seis caras, el estudiante gana 3 puntos, Calcula las respectivas probabilidades: \( p=0.5 \) \( n=10 \) a) \( P(X \geq 5)= \) b) \( P(X<3)= \) c) \( P(X>6)= \)

Question 1 (1 point) Choose 3 ways a sample space can be represented. (you will select 3 of these) Select 3 correct answer(s) List Taragraph Tree Diagram Equation

The Fundamental Counting Principle (FCP) store enlls three types of shirts: red, blue, and green. Each shirt is available three different sizes: small, medium, and large. a.) Draw a tree diagram that represents all the possible combinations of Use the tree diagram to calculate the total number of different shirt com- tions (color and size) available in the store (5 pts). Suppose a customer randomly selects one shirt from the store. What is robability that the selected shirt is cither blue or large in size? (5pts) What is the probability of a customer purchasing a shirt that is large irple? (5pts)

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.)

Question 3 (1 point) Flipping a coin once, then flipping it again is an example of: independent events dependent events

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.