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3 problems match “uniform distribution”. Clear

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1. Second and third rolls exceed first ●●○○○ 20 ptsA fair six-sided die is rolled three times. Find the probability that the second and third rolls are both larger than the first roll. Round to two decimal places.
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. Rebid probability for close bids ●●●○○ 50 ptsTwo construction companies make bids $X$ and $Y$ on a remodeling project. The joint density of $X$ and $Y$ is uniform on the rectangle $2<x<2.5$, $2<y<2.3$. If $X$ and $Y$ are within 0.1 of each other, the companies are asked to rebid; otherwise the lower bidder gets the contract. Find the probability that the companies are asked to rebid. Round to two decimal places.