Suppose Dan wins 25% of all checker games.

(a) What is the probability that Dan wins two checker games in a row?
(b) What is the probability that Dan wins four checker games in a row?
(c) When events are independent, their complements are independent as well. Use this result to determine the probability that Dan wins four checker games in a row, but does not win five in a row.

(a) The probability that Dan wins two checker games in a row is (Round to four decimal places as needed.)
(b) The probability that Dan wins four checker games in a row is (Round to four decimal places as needed.)
(c) The probability that Dan wins four checker games in a row but does not win five in a row is (Round to four decimal places as needed.)

(a) To find the probability that Dan wins two checker games in a row, we multiply the probability of winning a single game by itself since the events are independent.

The probability of winning a single game is 25%, or 0.25. Therefore, the probability that Dan wins two checker games in a row is:

0.25 * 0.25 = 0.0625

(b) Similarly, to find the probability that Dan wins four checker games in a row, we multiply the probability of winning a single game by itself four times:

0.25 * 0.25 * 0.25 * 0.25 = 0.00390625

(c) The probability that Dan wins four checker games in a row but does not win five in a row can be found by subtracting the probability of winning five games in a row from the probability of winning four games in a row. Since the events are independent, their complements are also independent. The complement of winning four games in a row is losing the fifth game. The probability of winning four games in a row is 0.00390625, and the probability of losing the fifth game is 0.75 (since Dan's chance of losing is 1 - 0.25). Therefore, the probability that Dan wins four games in a row but does not win five in a row is:

0.00390625 * 0.75 = 0.0029296875

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