Convert Biased Coin to Fair Coin
MediumGiven a biased coin function, write another function to make it return heads or tails with equal probability.
You are given a function biased_coin() that simulates the flip of a biased coin. The function returns 1 for heads and 0 for tails with unequal probabilities. Specifically, the probability of returning heads (1) is p, and the probability of returning tails (0) is 1 - p, where p is not necessarily 0.5.
Your task is to write a new function fair_coin() that uses biased_coin() to simulate a fair coin flip. The function fair_coin() should return 1 for heads and 0 for tails with equal probability (0.5 each).
Pythondef fair_coin() -> int:
pass
Requirements:
-
Use
biased_coin(): You must use thebiased_coin()function to implementfair_coin(). You cannot use any other random number generators or external libraries. -
Uniform Probability: Ensure that the
fair_coin()function returns1and0with exactly equal probability. -
Efficiency: Aim for an efficient implementation. While there is no strict constraint on the time complexity, avoid excessive or unnecessary computations.
-
Edge Cases: Consider edge cases such as very low or very high values of
p. Ensure that your solution handles these cases correctly.
The objective is to use the biased_coin() function to simulate a fair coin flip, where heads and tails are equally likely with a probability of 0.5 each. The solution exploits the fact that by flipping the biased coin twice, we can use the resulting pairs of outcomes to achieve fairness.
To implement this, the fair_coin() function repeatedly flips the biased coin two times. Each pair of outcomes from these flips can be one of four combinations: (0,0), (0,1), (1,0), or (1,1). When the two flips yield different outcomes, i.e., (0,1) or (1,0), this pair provides a fair result. For instance, if the outcome is (0,1), we return the result of the first flip, which is 0. Conversely, if the outcome is (1,0), we return the result of the first flip, which is 1. In cases where both flips yield the same result, either (0,0) or (1,1), the result is discarded, and the process is repeated.
This method ensures fairness because the pairs (0,1) and (1,0) occur with equal probability, given the bias in the coin flips. As a result, the fair_coin() function produces heads and tails with a probability of 0.5, effectively simulating a fair coin despite the bias in the individual coin flips.
Pythondef fair_coin() -> int:
while True:
flip1 = biased_coin()
flip2 = biased_coin()
if flip1 != flip2:
return flip1Related questions
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