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  3. 4 eight rooks are placed randomly on a chess board...

Question: 4 eight rooks are placed randomly on a chess board...

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4. Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? (In non-chess terms: Randomly pick 8 unit squares from an 8 x 8 square grid. What is the probability that no two squares share a row or a column?) Hint: How many choices do you have to place rooks in the first row? After you have made your choice, how many choices do you have for the second? Continue this reasoning We toss two dice. Consider the events 5. E: The sum of the outcomes is even. F: At least one out come is 5. Calculate the conditional probabilities P(E F) and P(FIE) 6. A fair die is rolled repeatedly (a) Give an expression for the probability that the first five rolls give a three at most two times (b) Calculate the probability that the first three does not appear before the fifth roll. (c) Calculate the probability that the first three appears before the twentieth roll, but not before the fifth roll. 7.* Let the sequence of events E1, E2,.... En be independent, and assume that P(E) Show that P(EUEn)

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