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2 problems match “circular permutations”. Clear

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1. Expected count of numbers exceeding a neighbor ●●●○○ 50 ptsThe numbers $1$ to $10$ are written in a circle in a random order. Find the expected number of numbers that are at least $2$ larger than at least one of their two adjacent numbers. Round to two decimal places.
2. Birthdays that are equal or adjacent ●●●●○ 100 ptsAssume 23 people have independent birthdays, each equally likely to be any of 365 days (February 29 is not considered), and January 1 and December 31 are considered adjacent. Find the probability that at least two people share the same birthday or have adjacent birthdays. Round to seven decimal places.