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1. Subsets with a gap pattern ●●○○○ 20 ptsDetermine the number of subsets $S$ of $\{1,2,3,\ldots,10\}$ with the following property: there exist integers $a<b<c$ with $a \in S$, $b \notin S$, $c \in S$.
2. Subsets contained in neither of two sets ●●○○○ 20 ptsFind the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$.
3. Birthday on a given day ●●○○○ 20 ptsA class has 30 students, and each student's birthday is equally likely to be any of 365 days. Find the probability that at least one student has a birthday on 5 November. Round to three decimal places.
4. Smallest group with a shared birthday ●●●○○ 50 ptsWhat is the smallest number of people in a group such that the probability that at least two of them share the same birthday is greater than 50%? Assume there are 365 equally likely birthdays and ignore leap years.
5. Rolling more 1s than 6s ●●●○○ 50 ptsIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1s than 6s? Round to two decimal places.
6. Ten numbered coins sum at least 45 ●●●○○ 50 ptsYou have 10 coins. Coin $i$ shows the number $i$ on one side (for $i = 1, \dots, 10$) and $0$ on the other side. You toss all the coins and add up the values that face up. Find the probability that the sum is greater than or equal to $45$. Give your answer as a decimal.
7. 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.