Ana and Becky want to play a safer version of Russian Roulette. The game starts with a coin toss. Whoever wins the toss puts a single round in a six-shot toy revolver, spins the cylinder, places the muzzle against a target, and pulls the trigger. If the loaded chamber aligns with the barrel, the weapon fires, and the player wins.
Given the rules of the game, how important is the toss?
We will evaluate the probability of each player – the one who wins (W) the toss and the one who loses (L).
Let’s write down the scenarios where the toss winner (W) wins the contest.
1) W wins in the first round
2) W doesn’t win in the first, survives the second (L’s chance) and wins the third.
3) W doesn’t win in the first, survives the second (L’s chance), doesn’t win the third, survives the fourth, and wins the fifth.
…
The total probability of W winning is the sum of all individual probabilities.
1) chance of winning the first round = 1/6
2) chance of winning the third round = chance of not winning the first x chance of not losing the second x chance of winning the third = (5/6)x(5/6)x(1/6) = (1/6)x(5/6)2
3) chance of winning the fifth = (1/6)x(5/6)4
Overall probability = 1/6 + (5/6)x(5/6)x(1/6) = (1/6)x(5/6)2 + (1/6)x(5/6)4 + (1/6)x(5/6)6 + … = 0.545 = 54.5%
On the other hand, the person who lost the toss has a total probability of 45.5% of winning the game. So, a 9% advantage is given by the toss.
Surviving The Deadliest 2-Player Game: Vsauce2
