How Bias Is Bias?

Probability Level 3

You want to maximize the odds of obtaining exactly 2 heads in a coin toss.

Would you rather toss 4 fair coins, or 4 identically biased coins?

Toss 4 biased coins There is insufficient information Toss 4 fair coins

1 solution

Zee Ell
Aug 20, 2016

If we assume, that all 4 coins are identical (either all fair or all of them are biased the very same way), then the probability of getting exactly 2 coins can be modeled by using the binomial distribution:

$Pr(X = 2) = {4 \choose 2} × p^2 × (1 - p)^2 = 6p^2(1- p)^2$ ,

where p is the probability of getting a head when tossing any (1) of the coins.

Due to the AM - GM inequality,

$p^2(1- p)^2$

is maximal when

$p = 1 - p \iff 2p = 1 \iff p = 0.5 ,$

which is the p value of a fair coin.

Therefore, in order to maximise our odds, we should:

$\boxed { \text {Toss 4 fair coins.} }$

A simpler way to maximize $f(p) = p^2 (1 - p)^2$ in the interval $0< p < 1$ is equivalent to maximizing the parabola $g(p) = \sqrt{f(p)} = p (1-p)$ . So there isn't a need to use AM GM.

Pi Han Goh - 4 years, 9 months ago

How Bias Is Bias?

1 solution

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