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Comparing dice rolls

Solve the first problem 16 problems

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1. Die roll greater than two ●○○○○ 10 ptsA fair six-sided die is rolled three times. The results are 1, 4, and 3, in that order. What is the probability that the next roll will produce a number greater than 2? Round to two decimal places.
2. Exactly two sixes in three rolls ●○○○○ 10 ptsLet $X$ be the number of sixes obtained in three rolls of a fair six-sided die, so $X \sim \text{Bin}(3, \tfrac{1}{6})$. Find $P(X = 2)$. Round to three decimal places.
3. Win probability of die A against die B ●●○○○ 20 ptsDie A has faces $3, 3, 3, 3, 3, 6$ and die B has faces $2, 2, 2, 5, 5, 5$, each face equally likely. Both dice are rolled at the same time. Find the probability that die A shows a higher value than die B. Round to two decimal places.
4. Win probability of die C against A ●●○○○ 20 ptsDie C has faces 1, 4, 4, 4, 4, 4 and die A has faces 3, 3, 3, 3, 3, 6, each face equally likely. Both dice are rolled at the same time. Find the probability that die C shows a higher value than die A. Round to two decimal places.
5. 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.
6. 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.
7. Non-decreasing rolls of a die ●●●○○ 50 ptsA fair die is tossed 6 times. What is the probability that on each toss, the number shown is greater than or equal to the number on the previous toss? Give your answer as a decimal rounded to four decimal places.
8. Ten ten-sided dice sum above 95 ●●●○○ 50 ptsTen fair dice are rolled together, each with equally likely faces numbered 1 through 10. Find the probability that the total of the ten top faces is greater than 95. Give your answer as a decimal.

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