The multiplication game

You walk into a casino, and just inside the main entrance you see a new game to play—the Multiplication Game. You sit at a table opposite the dealer and place your bet. The dealer hits a button and from a slot in the table comes a slip of paper with a number on it that you cannot see. You use a keypad to choose a number of your own—any positive integer you like, with as many digits as you like. Your number is printed on the slip of paper along with the product of the two numbers. The dealer shows you the slip so that you can verify that the product is correct. You win if the first digit of the product is 4 through 9; you lose if it is 1, 2, or 3. The casino pays even odds: for a winning bet of one dollar the casino returns your dollar and one more. Should you stay and play? It looks tempting. You you have six winning digits and the casino has only three! But being skeptical, you take a few minutes to calculate. You write the multiplication table of the digits from one to nine. Of the 81 products you see that 44 of them begin with 1, 2, or 3, and only 37 begin with 4 through 9. Suddenly, even odds do not seem so attractive! You abandon the game and walk further into the casino. In the next room you find another table with the same game, but better odds. This table pays $1.25 for a winning one dollar bet. From your previous count you figure that if the odds favor the casino by 44:37, then a fair payout would be 44/37 dollars for a dollar bet; that is almost $1.19, and this table is offering more. Should you stay and play? You open your laptop and write a computer program to count the products of the two digit numbers from 10 to 99. (You realize immediately that in this game multiplying by 1, 2, . . . , 9 is the same as multiplying by 10, 20, . . . , 90, and so you leave out the one digit numbers.) You find that of these 8100 products the casino has 4616 winners and you