At a restaurant you must choose one dessert out of three options with equal probability by using only a coin. How can this be done fairly and how can you still choose fairly if the coin is biased and the degree of bias is unknown?

Introduction / Context:
This puzzle combines probability with careful reasoning about randomness. Choosing one item out of three with equal probability using a coin is a standard interview question. The twist in the second part is that the coin may be biased in an unknown way, so you must use a method that converts this biased randomness into fair outcomes. Problems like this show your understanding of probability, independence, and rejection sampling ideas.

Given Data / Assumptions:
- There are exactly three desserts, which we can label A, B, and C.- You have only one coin and you can toss it repeatedly.- In the first scenario the coin is fair so heads and tails each have probability 1/2.- In the second scenario the coin may be biased but the direction and amount of bias are unknown.- The goal is to ensure that each dessert is chosen with the same probability in both scenarios.

Concept / Approach:
For a fair coin, the standard technique is to look at sequences of tosses. Two tosses give four equally likely outcomes HH, HT, TH, and TT. We can map three of them to the three desserts and reject the fourth by repeating the experiment whenever that fourth pattern occurs. For an unknown biased coin, a classical trick due to Von Neumann uses the pair HT as one fair outcome and the pair TH as another fair outcome and rejects HH and TT. That procedure extracts an unbiased virtual coin from a biased physical coin. We can then again build three equal outcomes from repeated fair virtual coin tosses or by using longer sequences.

Step-by-Step Solution:
Step 1: In the fair coin case toss the coin twice. The four equiprobable outcomes are HH, HT, TH, and TT.Step 2: Assign HH to dessert A, HT to dessert B, and TH to dessert C. If the result is TT reject this trial and toss the coin twice again until one of the first three patterns appears.Step 3: Each of HH, HT, and TH has probability 1/4 on any given pair of tosses. Because TT is simply discarded, the conditional probability that the chosen pair is HH, HT, or TH is 1/3 each, so A, B, and C are equally likely.Step 4: For the biased coin case, use the Von Neumann idea. Toss the coin twice and treat HT as a virtual fair heads and TH as a virtual fair tails. If the result is HH or TT discard that pair and toss again. Because the biased probabilities cancel out in the pair HT versus TH, this procedure simulates a fair coin.Step 5: Once you have a stream of virtual fair heads and tails from the previous step, you can again use pairs HH, HT, and TH with rejection of TT to map to desserts A, B, and C as in the fair coin case.

Verification / Alternative check:
The fairness can be checked with simple probability calculations. In the fair coin case, before rejection, each pair has probability 1/4. Conditioned on not getting TT, the probabilities of HH, HT, and TH all become (1/4) divided by (3/4), which is exactly 1/3. For the biased coin, let the probability of heads be p and tails be 1 minus p. Then probability of HT equals p times (1 minus p) and the probability of TH equals (1 minus p) times p, which are the same. So HT and TH are equally likely, independent of the value of p, and discarding HH and TT yields an unbiased virtual coin.

Why Other Options Are Wrong:
Option A gives dessert A probability 1/2 and the combination of desserts B and C probability 1/2, but B and C are not equally likely so the choice is not fair.Option C ignores the use of randomness entirely and therefore cannot guarantee equal chances.Option D is incorrect because the Von Neumann construction shows that you can obtain fair choices from an unknown biased coin by using pairs and rejecting some outcomes.

Common Pitfalls:
Many people try to assign two toss outcomes to one dessert and one outcome to another dessert without using rejection, which leads to unequal probabilities. Others believe that a biased coin can never produce fair results, forgetting that patterns of multiple tosses can cancel out the bias. A careful reading of the puzzle and a willingness to use repeated experiments are essential to reaching the correct method.

Final Answer:
Use two coin tosses and map HH, HT, and TH to the three desserts while repeating if TT appears, and in the biased coin case first convert pairs HT and TH into an unbiased virtual coin and then apply the same three way mapping.