Martin works at home. At 2 pm he decides to take a break to buy a copy of the Chroniclenewspaper. There are three nearby newsagents: Arif, Bob and Carol. However by 2 pm theymay have sold all their Chronicles and so have none available. The independent probabilitiesthat they have a Chronicle available at 2 pm are:Arif 0.4 Bob 0.7 Carol 0.25(a) State the probability that Bob does not have a Chronicle at 2 pm.(b) Find the probability that none of the three newsagents has a Chronicle available at 2pm.(c) (i) Find the probability that Bob does not have a Chronicle available at 2 pm but Arifdoes.(ii) Find the probability that Carol does not have a Chronicle available at 2 pm but Arifdoes.(d) Martin decides to visit newsagents in turn until he obtains a Chronicle or until he hasvisited all three. He tosses a coin. If it lands heads he will visit the three newsagents inthe order Bob, Arif, Carol. If it lands tails he will visit them in the order Carol, Arif,Bob.Find the probability that he will obtain a Chronicle from Arif.
Question
Martin works at home. At 2 pm he decides to take a break to buy a copy of the Chroniclenewspaper. There are three nearby newsagents: Arif, Bob and Carol. However by 2 pm theymay have sold all their Chronicles and so have none available. The independent probabilitiesthat they have a Chronicle available at 2 pm are:Arif 0.4 Bob 0.7 Carol 0.25(a) State the probability that Bob does not have a Chronicle at 2 pm.(b) Find the probability that none of the three newsagents has a Chronicle available at 2pm.(c) (i) Find the probability that Bob does not have a Chronicle available at 2 pm but Arifdoes.(ii) Find the probability that Carol does not have a Chronicle available at 2 pm but Arifdoes.(d) Martin decides to visit newsagents in turn until he obtains a Chronicle or until he hasvisited all three. He tosses a coin. If it lands heads he will visit the three newsagents inthe order Bob, Arif, Carol. If it lands tails he will visit them in the order Carol, Arif,Bob.Find the probability that he will obtain a Chronicle from Arif.
Solution
(a) The probability that Bob does not have a Chronicle at 2 pm is 1 - the probability that Bob does have a Chronicle at 2 pm. So, 1 - 0.7 = 0.3.
(b) The probability that none of the three newsagents has a Chronicle available at 2 pm is the product of the probabilities that each of them does not have a Chronicle. So, (1 - 0.4) * (1 - 0.7) * (1 - 0.25) = 0.6 * 0.3 * 0.75 = 0.135.
(c) (i) The probability that Bob does not have a Chronicle available at 2 pm but Arif does is the product of the probability that Bob does not have a Chronicle and the probability that Arif does. So, 0.3 * 0.4 = 0.12. (ii) The probability that Carol does not have a Chronicle available at 2 pm but Arif does is the product of the probability that Carol does not have a Chronicle and the probability that Arif does. So, 0.75 * 0.4 = 0.3.
(d) The probability that he will obtain a Chronicle from Arif depends on the order in which he visits the newsagents. If he visits Bob first, the probability is 0.3 * 0.4 = 0.12. If he visits Carol first, the probability is 0.75 * 0.4 = 0.3. Since he decides the order by tossing a coin, the probabilities are equally likely. So, the overall probability is (0.12 + 0.3) / 2 = 0.21.
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