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Q1:

Harry is an amateur birdwatcher who goes birdwatching in a region in which he knows there are only two species of thornbill – the buff-rumped thornbill and the yellow-rumped thornbill. Harry has some difficulty in differentiating the two species (but he does know a thornbill when he sees one). If Harry sees a buff-rumped thornbill, he will correctly identify it 80% of the time. If he sees a yellow-rumped thornbill, he will correctly identify it 60% of the time. In this region, 70% of the thornbills are buff-rumped.

(i) If Harry sees a thornbill, what is the probability that he thinks it is a yellow-rumped?

Harry goes birdwatching in this region with George who is a thornbill expert. If Harry correctly identifies a thornbill, George will correctly identify it 99% of the time. If Harry misidentifies a thornbill, George will correctly identify it 90% of the time.

(ii) Harry and George both spot a thornbill which they both identify as a yellow-rumped. What is the probability that the bird is in fact buff-rumped?

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Q2:

At the start of each month, Microsoft classifies Xbox Live users either as Gold, Silver or Bronze members. Gold members have a 70% chance of staying a Gold member, and can never become Bronze members by the end of the next month. If a user is a Silver member at the start of a given month, then they are 3 times as likely as becoming a Gold member compared with a Bronze member at the start of the next month. They also have a 40% chance of staying a Silver member.

Finally, 50% of Bronze members remain Bronze members, while 50% become Silver members at the start of the next month. For the 50% of Bronze members who would become Silver members at the start of the next month, Microsoft randomly offers 10% of these users a chance to become a Gold member at the start of the next months (hence it is possible for Bronze members to become Gold members in one month). To accept this offer, the user must pay a small fee, so not everybody accepts. Microsoft estimates that 90% of Bronze members accept this offer.

(a) Set up the probability transition matrix for this Markov chain, with a user’s membership considered at the beginning of each month.

(b) If a user is a Bronze member at the beginning of April, what is the probability that they will be a Gold member at the beginning of June?

(c) Determine the proportion of users who are Gold members in the long run.

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# Homework Help: Draw a tree of possibilities

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