This week the #puzzle is: How Low (or High) Can You Go? #probabilities #volume #area #random

You’re playing a game of “high-low,” which proceeds as follows:

First, you are presented with a random number, x1, which is between 0 and 1.

A new number, x2, is about to be randomly selected between 0 and 1, independent of the first number. But before it’s selected, you must guess how x2 will compare to x1. If you think x2 will be greater than x1 you guess “high.” If you think x2 will be less than x1, you guess “low.” If you guess correctly, you earn a point and advance to the next round. Otherwise, the game is over.

If you correctly guessed how x2 compared to x1 then another random number, x3, will be selected between 0 and 1. This time, you must compare x3 to x2, guessing whether it will be “high” or “low.” If you guess correctly, you earn a point and advance to the next round. Otherwise, the game is over.

You continue playing as many rounds as you can, as long as you keep guessing correctly.

You quickly realize that the best strategy is to guess “high” whenever the previous number is less than 0.5, and “low” whenever the previous number is greater than 0.5.

With this strategy, what is the probability you will earn at least two points? That is, what are your chances of correctly comparing x2 to x1 and then also correctly comparing x3 to x2?

And for extra credit:

Your friend is playing an epic game of “high-low” and has made it incredibly far, having racked up a huge number of points.

Given this information, and only this information, what is the probability that your friend wins the next round of the game?

How Low (or High) Can You Go?

Highlight to reveal (possibly incorrect) solution:

Desmos

This problem can be modelled as the volume of a shape. Imagine a unit cube. The good parts of the volume are where:

• x was low, x < y, y was low, y < z. E.g. (0.1, 0.2, 0.7).
• And the 3 other variations of x and y being high and low.

See Desmos for a calculation of the volume: 13/24 = 0.54167.

And for extra credit:

Program Spreadsheet

As there is no history in probabilities, I simply need to look at 2 guesses in a row. So this time we’re looking at the good part of the area of a unit square. The first guess can be anything. See Desmos for calculation of the area: 0.75.

Just to make absolutely sure, I wrote a small program to simulate winning a lot of times in a row. Behaves as expected.

Alternate answer: My friend is cheating. The probability is 100%.

I decided to take another look at the data. Wrote a better program. Among other things, it produced data for a spreadsheet. What do you know! The data skews somewhat. There’s a good chance, the latest number drawn was in the middle, was close to 0.5. This has an effect on whether my next guess will be correct. My program says:

Probability for win at length 10: 0.72113
Probability for win at length 20: 0.72152
Probability for win at length 30: 0.71378

So instead of 0.75 (or 1), it appears the probability is 0.71 or 0.72.