The lucky derby

In the spirit of the Kentucky Derby, this Riddler puzzle is about a peculiar type of horse race.

The bugle sounds, and 20 horses make their way to the starting gate for the first annual Lucky Derby. These horses, all trained at the mysterious Riddler Stables, are special. Each second, every Riddler-trained horse takes one step. Each step is exactly one meter long. But what these horses exhibit in precision, they lack in sense of direction. Most of the time, their steps are forward (toward the finish line) but the rest of the time they are backward (away from the finish line). As an avid fan of the Lucky Derby, you’ve done exhaustive research on these 20 competitors. You know that Horse One goes forward 52 percent of the time, Horse Two 54 percent of the time, Horse Three 56 percent, and so on, up to the favorite filly, Horse Twenty, who steps forward 90 percent of the time. The horses’ steps are taken independently of one another, and the finish line is 200 meters from the starting gate.

Handicap this race and place your bets! In other words, what are the odds (a percentage is fine) that each horse wins?

Here is my full derivation (long!):
[Show Solution]

For the tl;dr, the answer is:
[Show Solution]

The deadly board game

This Riddler classic puzzle involves a combination of decision-making and probability.

While traveling in the Kingdom of Arbitraria, you are accused of a heinous crime. Arbitraria decides who’s guilty or innocent not through a court system, but a board game. It’s played on a simple board: a track with sequential spaces numbered from 0 to 1,000. The zero space is marked “start,” and your token is placed on it. You are handed a fair six-sided die and three coins. You are allowed to place the coins on three different (nonzero) spaces. Once placed, the coins may not be moved.

After placing the three coins, you roll the die and move your token forward the appropriate number of spaces. If, after moving the token, it lands on a space with a coin on it, you are freed. If not, you roll again and continue moving forward. If your token passes all three coins without landing on one, you are executed. On which three spaces should you place the coins to maximize your chances of survival?

Extra credit: Suppose there’s an additional rule that you cannot place the coins on adjacent spaces. What is the ideal placement now? What about the worst squares — where should you place your coins if you’re making a play for martyrdom?

Here is my solution:
[Show Solution]

Random walk with endpoints

The Riddler post for today is about a bar game where you flip a coin and move forward or backward depending on the result. Here is the problem:

Consider a hot new bar game. It’s played with a coin, between you and a friend, on a number line stretching from negative infinity to positive infinity. (It’s a very, very long bar.) You are assigned a winning number, the negative integer -X, and your friend is assigned his own winning number, a positive integer, +Y. A marker is placed at zero on the number line. Then the coin is repeatedly flipped. Every time the coin lands heads, the marker is moved one integer in a positive direction. Every time the coin lands tails, the marker moves one integer in a negative direction. You win if the coin reaches -X first, while your friend wins if the coin reaches +Y first. (Winner keeps the coin.)

How long can you expect to sit, flipping a coin, at the bar? Put another way, what is the expected number of coin flips in a complete game?

Here is my solution:
[Show Solution]

and here is a much slicker solution, courtesy of Daniel Ross:
[Show Solution]