Instructions: You must compose your assignments independently; however, you may discuss your work with one another...

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Instructions: You must compose your assignments independently; however, you may discuss your work with one another at the rough level. 1. (5 points) Show that if A and B are independent, then A and B are also independent. 2. (10 points) Let A and B be two independent events with P(A) = 0.4 and P(AUB) = 0.64. What is P(B)? 3. (10 points) Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the 148 students is randomly selected. Let X denote the number of students who were on the bus carrying this randomly selected student. Independently, one of the 4 bus drivers is also randomly selected. Let Y denote the number of students on his/her bus. (a) Find the distribution of X. (b) Find the distribution of Y. 4. (5 points) Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that the subsystem (of components 1 and 2) works if and only if either 1 or 2 works. Since 3 and 4 are connected in series, that subsystem (of components 3 and 4) works if and only if both 3 and 4 work. If each components works with probability 0.9, independently of others. Calculate the probability that the whole system works. 4 Instructions: You must compose your assignments independently; however, you may discuss your work with one another at the rough level. 1. (5 points) Show that if A and B are independent, then A and B are also independent. 2. (10 points) Let A and B be two independent events with P(A) = 0.4 and P(AUB) = 0.64. What is P(B)? 3. (10 points) Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the 148 students is randomly selected. Let X denote the number of students who were on the bus carrying this randomly selected student. Independently, one of the 4 bus drivers is also randomly selected. Let Y denote the number of students on his/her bus. (a) Find the distribution of X. (b) Find the distribution of Y. 4. (5 points) Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that the subsystem (of components 1 and 2) works if and only if either 1 or 2 works. Since 3 and 4 are connected in series, that subsystem (of components 3 and 4) works if and only if both 3 and 4 work. If each components works with probability 0.9, independently of others. Calculate the probability that the whole system works. 4