{a} A standard adult football match consists of two periods of 45 minutes each. Sup- pose that, in a certain football competition, the time between two consecutive goals follows an exponential distribution with a mean of 20 minutes. Consider a match starts at 12:08 noon. 1. Find the probability that less than 2 goals will be scored in rst 45 minutes. [3 marks} ii. Find the probability that it takes less than 10 minutes to score the rst goal. [3 marks] iii. Find the expected number of goals will he soared in the second half of the match. [2 marks] iv. Suppose that the nal score is 1-]. Find the probability the rst goal is scored in the rst 20 minutes of the match. [4 marks] v. The match will go into extra-time when two teams score the same number of goals in the regular time of a match. The extra-time is an additional 30- minute period after the match. Find the probability that a goal is scored during the extra-time. given that neither of the teams has scored a goal in the regular time of the match. [3 marks] (1:) Mandy has just posted a photo in her social media account. Her ioowera write comments on the photo and the inter~arriml time of the comments follows an exponential distribution with a mean of 15 minutes. Mandyr wants to reply each of the comments on her own and she can write on average 6 replies in an hour. Suppose that the comments are replied on e rst-oome-rstBerved basis. Determine the following: i. The steadyetate probability that there is no unreplied comment. [3 marks] ii. The expected number of unreplied comments (including the comment cur- rently being replied by Mandy). [3 marks] iii. The expected waiting time of a. certain comment to he replied [excluding the time for Mandy to write the reply for this comment). [4 marks]