Photo:
THE MIRROR
The rules of the numbers game are simple: a random number generator rolls two numbers that lie in the interval from 0 to 1.
Contrary to what the picture above suggests, the number can have hundreds of decimal places.
The probability that both random numbers are the same is so small that we can ignore this case.
The first number is displayed and Paul then has to decide whether the second number, which he does not yet know, is larger or smaller.
If he is correct, he wins this round and gets a point.
If his choice is wrong, his teammate Martha gets the point.
Paul uses a simple and apparently successful strategy: If the displayed number is greater than (or equal to) 0.5, he bets that the second number is smaller.
If the number displayed is less than 0.5, his tip is: number two is greater.
Martha and Paul play for a while, but Paul wins a lot more often.
To make the game more exciting, the two change the rules.
Martha can look at the two random numbers in advance and select the one that is shown to Paul.
What must Martha do to increase her chances of winning?
What is the probability of her winning now?
What is the probability that Paul won the game using the original rules?
Martha can increase her chances of winning to
50 percent
by using the strategy outlined below.
In the original version of the game, Paul has a
75 percent
chance of winning .
First to the original version: A random number is less than or greater than (or equal to) 0.5 with a probability of 50 percent.
Since two random numbers are drawn, there are four possibilities, each with a probability of 25 percent:
Number 1 is less than 0.5 and number 2 is greater than or equal to 0.5 (Paul always wins)
Number 1 is greater than or equal to 0.5 and number 2 is less than 0.5 (Paul always wins)
Numbers 1 and 2 are less than 0.5 (Paul has a 50 percent chance of winning)
Numbers 1 and 2 are greater than or equal to 0.5 (Paul has a 50 percent chance of winning)
If we add up the four cases, we get 75 percent.
How can Martha increase your chances of winning to 50 percent?
From the two numbers, she chooses the one that has the smallest distance to 0.5.
She shows this number to Paul.
If the number chosen has a distance of a from 0.5, then the second number is either in the interval from 0 to 0.5-a or in the interval from 0.5+a to 1. In the first case, the second number is smaller, in the second case it is larger.
Paul cannot know which of the two cases applies.
His previous strategy no longer works.
Because there is a 50 percent chance that the second number is greater than or less than Martha's choice, regardless of whether Martha's choice is greater than or equal to or less than 0.5.
Even if Paul saw through Martha's strategy, that doesn't help him any further.
I discovered this riddle in Peter Winkler's book »Mathematical Riddles for Lovers«.
In case you missed a mystery from the past few weeks, here are the most recent episodes:
How long is the red line?
Does a cube fit through itself?
The monster number and its two non-divisors
sink squares
Good luck and bad luck in the raffle
heirs in the square
The super heavy freight train
Magic with prime numbers
Strict logic in the rabbit hutch
You shall be many friends