In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and winning requires that your sequence of four digits matches the four digits that are later drawn. Assume you placed one bet with a sequence of four digits.
a. Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find P (MMXX), where M denotes a match and X denotes a digit that does not match the winning number.
b. Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list.
c. Based on the preceding results, what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?
The solution comprises of detailed step-by-step calculations using the multiplication rule to find the probability of winning the lottery.