Biased Coin

Concise Answer (continued): Yes, use a sequence of flips to simulate a fair outcome.
Detailed Explanation (continued): For an unfair coin with any bias, you can flip it multiple times to create pairs of outcomes such as HT and TH. If the coin lands HT, you decide “heads,” and if it lands TH, you decide “tails.” If it lands HH or TT, you disregard the outcome and flip again. This method effectively ‘cancels out’ the bias, as HT and TH have the same probability, and you keep flipping until you get one of these combinations. This process ensures a 50% chance for each outcome and is known as the von Neumann extractor for fairness.
Now, let’s add more detail for parts b), c), and d) since these require a deeper explanation:

b) Expected Number of Flips (Detailed): The expected number of flips until the realization of an event is the inverse of the event’s probability. If the event is “getting heads” with P(H) = 1/3, the expected number of flips is 1/(1/3), which equals 3 flips. For two consecutive heads, we need to consider the geometric series formed by the probabilities of getting two heads in a row for each series of independent flips. The probability of getting two heads in a row is (1/3)^2 = 1/9, so the expected number of flips would be 1/(1/9) = 9 flips. However, because any flip sequence ending in “TH” resets the count for two consecutive heads, the calculation is a bit more complex, resulting in an expectation of 12 flips.

c) Reducing the Number of Flips (Concise): No strategy can change the expected number of flips because the coin’s bias is constant, and each flip is independent.

c) Reducing the Number of Flips (Detailed): Since the outcomes are determined by the biased probability of the coin, the expected number of flips for a given event cannot be changed without changing the underlying probability. However, what can be done is optimizing the process by avoiding unnecessary flips once the desired event has occurred or by using outcomes in combination to decide an event (as in the HT/TH method).

d) Strategy for Any Unfair Coin (Concise): The HT/TH method can be adapted for any bias.

d) Strategy for Any Unfair Coin (Detailed): The general approach to create a fair outcome from a biased coin is to flip the coin twice, creating four outcomes: HH, HT, TH, and TT. For any bias, the outcomes HT and TH will have the same probability, which allows us to use them to simulate a fair coin. If neither of these outcomes occurs, the flips are repeated. This strategy ensures a fair event from a biased coin with any probability distribution, as long as the coin is not deterministic (i.e., P(H) != 0 or 1).