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I am a bit confused on the way to go for this exercise could you please help me :

You have five coins in your pocket. It is known a priori that one coin has heads with probability 0.4 and the other four coins give heads with probability 0.7

A coin is being pulled out one of the five coins at random from your pocket (each coin has probability $\frac{1}{5}$ of being pulled out), and you want to find out which of the two types of coin it is. To that end, you flip the coin 6 times and record the results of each coin flip where $ X_1 = 1 $ if “heads" and $ X_1 = 0 $ if “tails".

Suppose that $ X_1 +X_2+X_3+X_4+X_5+X_6 = 3 $. Find the maximum likelihood estimate $\hat{p}^{MLE}$ for using the given data.(Enter a numerical value accurate to 3 decimal places.)

My first issue is that how can i find a single p mle when I have 2 probabilities for head. I proceeded as follows :

$$\prod_{i = 1}^{6} \left[\frac{1}{5} p^{x_i} (1-p)^{1-x_i} + \frac{4}{5} p^{x_i} (1-p)^{1-x_i}\right]$$

The $x_i$ will be replaced by 3, 1 will become $n$ i.e 6, which then becomes 3..

Then I will differentiate with respect to $p$ which will give me $0.5$ and this is wrong.

Could you please help me on that?

Thank you

SB

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The probability of observing $3$ heads from $6$ flips is $20 p^3 (1-p)^3$. There are two possible values of $p$:

  • when $p=0.4$ this is $0.27648$
  • when $p=0.7$ this is $0.18522$

so the maximum likelihood estimate corresponds to the higher likelihood and would be $\hat{p}=0.4$


That calculation took no account of the fact that there were originally one coin with $p=0.4$ compared to four coins with $p=0.7$. If you had, then the more likely value for $p$ would have $0.7$, found by considering the posterior probability that the flipped coin had $p=0.4$, a Bayesian calculation you were not asked for.

If you had been, then the answer would be $\dfrac{\frac15 0.27648}{\frac15 0.27648 + \frac45 0.18522}= \dfrac{128}{471} \approx 0.272$, which is higher than the original $\frac15$ but lower than the posterior probability of about $0.728$ that the flipped coin had $p=0.7$

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    $\begingroup$ in the question, it's been asked that the answer should be given with 3 decimal places, with 0.4 an answer, I tend to think that may be its not the correct one? $\endgroup$
    – Sitingbull
    May 4 '19 at 10:28
  • $\begingroup$ Siting: There are only two possible reasonable answers: $p=0.4$ or $p=0.7.$ $\endgroup$
    – whuber
    May 4 '19 at 22:35

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