A dice is handed out to each the students along with the test1. An urn contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.
How many balls are in the urn? (Before the extra balls are added.)
2. An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained
n matches, what is the probability that the other box currently contains k matches? (Where 0 < k < n.)
3. Roll a standard pair of six-sided dice, and note the sum. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:
a. The set of dots on each die is not the standard {1,2,3,4,5,6}.
b. Each face has at least one dot.
c. The number of ways of obtaining each sum is the same as for the standard dice.
4. Is the number 2438100000001 prime or composite?
5. What is the expected number of times a fair die must be thrown until all scores appear at least once?
6. A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in length. What is the maximum area he can enclose by connecting the pieces? Assume the land is flat.
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