The Sum of our Knowledge

This problem, along with a couple of others, was inspired by the work of John H. Conway* and appeared in  the October 15, 2020 edition of Quanta Magazine . Let a, b, c, d, e, f, g, h, i, j be the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in some order. Each digit appears exactly once. (For those that want a more precise mathematical phrasing, there is a one-to-one correspondence between the set {a, b, c, d, e, f, g, h, i, j} and the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, or better yet, there is a bijection between the two sets.)  We form other integers by concatenating digits, so for example, 'ab' is a two-digit number where 'a' is the tens digit and 'b' is the ones digit. If abcdefghij is a 10-digit number with the following properties: a is divisible by 1 (OK, that much is obvious) ab is divisible by 2 (So, b must be even) abc is divisible by 3 (Do you know the test for divisibility by 3?) abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg ...

COME ON DOWWWWNNNN!!!!!!   You’ve been chosen to be a contestant on Let’s Make a Deal!   You’re now standing in front of three doors -- behind one of these doors is a car, and behind each of the other two is a goat. Monty, the host, says, “Pick a door! If it has the car behind it, you’ll drive it home today!”  You choose door #1, and feel really good about your choice.  The host then opens door #3,  which has a goat behind it, and asks “Do you want to switch doors?”  Question: What is the probability that you will win the car now that you know there was a goat behind door #3? Should you switch? Assumptions: Before the show, the car was equally likely to be placed behind any of the three doors. Once you make your selection of door #1, Monty will choose to reveal what is behind one of the other two doors, but will definitely not show you where the car is. Update: 11/19/20: read below for the solution.