Exercise 2.1 (Sample Space and Events) The rise times (unit: min.) of a reactor for two batches are measured in an experiment. 1. Define the sample space of the experiment. 2. Define Er where the reactor rise time of the first batch is less than 55 min. and E2 where the reactor rise time of the second batch is greater than 70 min. 3. Find Ei U Ez, Ei n Ez, and Er'. 4. Are Ei and Ez mutually exclusive? 5. Are Er and Ez exhaustive? Engr. A. CUH-ING Exercise 2.2 (Probability of Joint Event) Test results of scratch resistance and shock resistance for 100 disks of polycarbonate plastic are as follows: Shock Resistance High (B) Low (B') Scratch High (A) 80 9 Resistance Low (A') 6 5 Let A and A' denote the event that a disk has high scratch resistance and the event that a disk has low scratch resistance, respectively. Let B and B' denote the event that a disk has high shock resistance and the event that a disk has low shock resistance, respectively. 1. When a disk is selected at random, find the probability that both the scratch and shock resistances of the disk are high. 2. When a disk is selected at random, find the probability that the scratch orshock resistance of the disk is high. 3. Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive? Exercise 2.3 (Conditional Probability) In Exercise 2.2, find the probability that a disk has high scratch resistance (A) when it has high shock resistance (B). Exercise 2.4 (Multiplication and Total Probability Rule) Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products, 60% of moderately successful products, and 10% of poor products received good reviews. In addition, 40% of product designs have been highly successful, 35% have been moderately successful, and 25% have been poor products. Find the probability that a product receives a good review. Engr. A. CUH-ING Exercise 2.5 (Independence) For the resistance test results in Exercise 2.2, the following probabilities have been calculated: 40 P(A) = 89 - and P(A | B) = 100 43 Assess if events A and B are independent. Exercise 2.6 (Bayes' Theorem) In Exercise 2.4, what is the probability that a new design will be highly successful if it receives a good review