Making a Biased Penny Fair

· 2 min read

Definitions

Fair penny: This type of penny has a 50/50 chance of landing on heads or tails. It’s fair.

Biased penny: A penny that’s not fair.

Assumptions:

  1. Each flip is independent of each other.
  2. The bias level of getting heads cannot be 1 or 0. (i.e. Always getting heads or never getting heads)

Now how can we make a biased penny fair? The solution is actually very elegant and simple:

Flip it twice

If you flip a penny twice, here are your outcomes:

  1. Heads, Heads
  2. Heads, Tails
  3. Tails, Heads
  4. Tails, Tails

Suppose your coin was biased, say it had a 60% chance of landing on Heads, and 40% of landing on Tails. (We’ll assume it’s impossible to land on its side)

Then the probability of each event:

  1. Pr[HH] = 0.6 * 0.6 = 0.36
  2. Pr[HT] = 0.6 * 0.4 = 0.24
  3. Pr[TH] = 0.4 * 0.6 = 0.24
  4. Pr[TT] = 0.4 * 0.4 = 0.16

Notice that Pr[HT] == Pr[TH], in other words the probability of a double coin flip landing on Heads followed by Tails is the same as the probability of a flip landing on Tails followed by a flip landing on Heads.

We’ll use this fact to make the penny fair.

Now we can make this penny fair by flipping it twice, then viewing its outcome.

  • If the outcome was HH or TT, then we’ll discard these results.
  • If the outcome was HT or TH, then we’ll take the result of our first flip.

Using Python to show this

from collections import namedtuple
import random

Coin = namedtuple('Coin', ['Heads', 'Tails'])
coin = Coin(Heads=.60, Tails=.40) # Our biased coin.

def flip(prob_head):
 return 'Heads' if random.random() < prob_head else 'Tails'

# Confirm that our coin is biased.
simulations = 1000000
flips = [flip(coin.Heads) for _ in range(simulations)]

pHead = len([flip for flip in flips if flip == 'Heads'])
print("Probability of Head with Biased Coin", pHead / len(flips))
# Probability of Head with Biased Coin 0.60039

# Let's make our coin fair!
flips = []
for _ in range(simulations):
 first, second = flip(coin.Heads), flip(coin.Heads)
 if first != second:
  flips.append(first)

pHead = len([flip for flip in flips if flip == 'Heads'])
print("Probability of Head with Biased Coin", pHead / len(flips))
# Probability of Head with Biased Coin 0.49952640042293545

Pretty cool huh?