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4 problems match “expected value”. Clear

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1. Expected consecutive pairs in a subset ●●○○○ 20 ptsA 25-element subset is chosen uniformly at random from $\{1,2,\ldots,100\}$. Find the expected number of pairs $(i,i+1)$ of consecutive integers for which both $i$ and $i+1$ lie in the subset.
2. Optimal bid for an object ●●●○○ 50 ptsThe owner's value $V$ of an object is uniformly distributed on $[0, 100]$ dollars, and its value to you is $1.8V$. You make one sealed bid of $b$ dollars. If $b$ is more than the owner's value, he sells it to you and you pay $b$; otherwise you get nothing. Choose $b$ to maximise your expected profit. What bid $b$ (in dollars) should you make?
3. Draws to see all six unique balls ●●●○○ 50 ptsA jar contains 66 white balls and 6 unique balls: 1 red, 1 blue, 1 green, 1 purple, 1 yellow and 1 cyan. Balls are drawn one at a time, each ball is returned to the jar after it is drawn, and the colours are recorded. What is the expected number of draws needed until all 6 uniquely coloured balls have been seen at least once?
4. Expected people until first birthday match ●●●●○ 100 ptsAssume birthdays are independent and uniformly distributed over 365 days. People are questioned one at a time. Find the expected number of people questioned until the first person shares a birthday with someone already questioned (this person is counted). Round to two decimal places.