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Data Interview Question

Balancing an Unbiased Coin

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Solution & Explanation

To transform an unfair coin into a fair one, we can utilize a method known as Von Neumann's Trick. This method involves flipping the unfair coin twice and interpreting the outcomes in a specific way to simulate a fair coin. Here's how it works:

Step-by-Step Process:

  1. Identify the Outcomes:

    • When you flip a coin twice, there are four possible outcomes:
      • HH (Head-Head)
      • TT (Tail-Tail)
      • HT (Head-Tail)
      • TH (Tail-Head)

  2. Select Fair Outcomes:

    • Among these outcomes, HT and TH have equal probability, even if the coin is biased. This is because the probability of HT is P(H) * P(T) and the probability of TH is P(T) * P(H), both of which are equal.

  3. Discard Unfair Outcomes:

    • Discard the outcomes HH and TT because they do not provide an unbiased result.

  4. Assign Fair Results:

    • If the outcome is HT, consider it as a "Head".
    • If the outcome is TH, consider it as a "Tail".

  5. Repeat Until Successful:

    • If you get HH or TT, discard the result and flip the coin twice again until you achieve either HT or TH.

Explanation:

  • Why HT and TH?

    • The key to this method is recognizing that the probability of HT and TH are equal, regardless of the bias of the coin. This is because both sequences involve one head and one tail, just in different orders.

  • Why Discard HH and TT?

    • The outcomes HH and TT are discarded because they don't help us achieve a fair result. They would skew the results towards heads or tails depending on the bias of the coin.

  • Efficiency:

    • While this method can be inefficient because you might need to flip the coin multiple times to get HT or TH, it guarantees that the result will be fair when you do.

Practical Example:

Suppose you have a coin where the probability of heads (H) is 0.7 and tails (T) is 0.3. If you flip it twice:

  • Probability of HT = 0.7 * 0.3 = 0.21
  • Probability of TH = 0.3 * 0.7 = 0.21

Both HT and TH have the same probability, making them suitable for simulating a fair coin.

This method provides a simple yet effective way to simulate a fair coin using an unfair one, ensuring that each outcome (heads or tails) has an equal chance of occurring.