Probaility

(a) 7 points Below is a 4 X 4 grid consisting of a total of 16 individual squares. On the right is a picture of a red square stamp. A child is playing with the stamp and and decides to stamp the squares as follows: for every individual square in the grid she rolls a die and if the die is an even number, she stamps that square. Notice that the maximum number of stamps she could make is 16. Depending on how she carries out the stamps, some of the line segments will be colored red. In the example below, she stamped 3 squares and there are 11 line segments that are colored red. Based on the child's game, answer the following: o What is the probability that she stamps exactly 2 squares? e What is the expected number of squares that she stamps? e What is the expected number of line segments that are colored red? 0 Is the number of red line segments a Binomial random variable? (c) 6 points Ten empty buckets are lined up in a row. Each bucket is labelled. A set of as balls are randomly thrown one after the other into the buckets. o What is the expected number of throws that are necessary until two buckets have at least one ball in them? For example, if the rst ball landed in bucket 1, and the second ball landed in bucket 2, then it only took me 2 throws. If the rst ball landed in bucket 1 and the second ball landed in bucket 1, and the third ball landed in bucket 4, then it took me 3 throws. a After all 11 balls are thrown, what is the probability that the total number of balls in bucket 1 and bucket 2 is 4? (Not 4 balls in each. A total of 4 balls in bucket 1 and bucket :2)