18. (This question is worth 12 points) A shopping mall is holding a promotional lottery. After purchasing goods of a certain amount, customers can participate in a lottery. Each lottery involves randomly drawing one ball from box A (containing 4 red balls and 6 white balls) and one ball from box B (containing 5 red balls and 5 white balls). If both balls drawn are red, the customer wins the first prize; if exactly one ball is red, the customer wins the second prize; if neither ball is red, the customer wins no prize. (1) Find the probability that a customer wins a prize in one lottery; (2) If a customer has 3 lottery chances, let X denote the number of times the customer wins the first prize in the 3 lotteries. Find the probability distribution and mathematical expectation of X.
18. (This question is worth 12 points)\\
A shopping mall is holding a promotional lottery. After purchasing goods of a certain amount, customers can participate in a lottery. Each lottery involves randomly drawing one ball from box A (containing 4 red balls and 6 white balls) and one ball from box B (containing 5 red balls and 5 white balls). If both balls drawn are red, the customer wins the first prize; if exactly one ball is red, the customer wins the second prize; if neither ball is red, the customer wins no prize.\\
(1) Find the probability that a customer wins a prize in one lottery;\\
(2) If a customer has 3 lottery chances, let X denote the number of times the customer wins the first prize in the 3 lotteries. Find the probability distribution and mathematical expectation of X.