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Say you have a coin A that has probability of $\theta$ of landing on heads and a coin B with probability of $2\theta$ of landing heads. Then say we flipped A 7 times and the first 5 flips were tails and the last 2 were heads. Then we flipped B 3 times where the first 2 were heads and the last was tails.

The likelihood function (Not totally sure if I did this right):
$P(Results|\theta) = \theta^2(1-\theta)^5\times(2\theta)^2(1-2\theta)$

The MLE function can be written as:
$log(P(Results|\theta)) = 2log(\theta)+5log(1-\theta)+2log(2\theta)+log(1-2\theta)$
$$\frac{\partial l(\theta)}{\partial \theta} = \frac{2}{\theta}-\frac{5}{1-\theta}+\frac{4}{2\theta}-\frac{2}{1-2\theta}=0$$

Did do this correctly? How can I write the last equation as a function of $\theta$

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$$\frac4\theta=\frac{5}{1-\theta}+\frac2{1-2\theta}$$

$$4(1-\theta)(1-2\theta)=5\theta(1-2\theta)+2\theta(1-\theta)$$

$$8\theta^2-12\theta+4=-12\theta^2+7\theta$$

$$20\theta^2-19\theta+4=0$$

To find $\theta$, you just have to solve the quadratic equation and note that since $2\theta \le 1$, we have $\theta \le \frac12$.

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