Hello, can you please help me with the questions? Please write the intermediate steps and clearly indicate the answers. Thank you.

Exercise 1. A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability function for the number of red lights she hits (X), as shown below: 0 1 2 3 4 5 P(X = x) P 0.21 10p 0.12 0.15 2p a) Find p and compute the probability that she hits at most 3 red lights. (2pts) b) What is the chance that she has hit at most 4 red lights, given that she has hit more than 1 red lights? (2pts) c) Compute the probability that she has hit at most 4 red lights, given that she has hit at least 1 red lights? (2pts) d) In a 5 workday week, what is the chance that she will hit more than 3 red lights in a single day at least on two days? You may assume that the traffic lights' behaviour are independent from day to day. (4pts) Exercise 2. An appliance dealer sells 5 types of air conditioner, namely 1 ton, 2 tons, 5 tons, 10 tons and 20 tons. Let X denote the tonnage of the air conditioner purchased by a random customer. Based on previous year's purchase records, the following probability distribution table is obtained: (in tons) 1 2 5 10 20 P(X = =) 0.08 0.18 0.24 0.36 0.14 a) Compute (i) E(X), (ii) E(X'), and (iii) using the values of (i) & (ii) compute Var(X). (2+2+1= 5pts) b) The air conditioners are priced by the retailer at 200 per ton. The retailer pays E120 per ton to buy the air conditioners. In addition there is an overhead cost of E35 per air conditioner. What is the expected value standard deviation of the profit for the retailer from the sale of an air conditioner? (5pts)