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We have two coins one fair and the other biased for which the probability of head is 2/3. We choose a coin at random, and we flip it two times. What is the probability that in both flips we will receive identical results? Obviously i know how to find single events probabilities however i do not know how to combine everything to get a desired result.

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  • $\begingroup$ PS: the terms are "fair" and "biased" not "correct" or "incorrect". $\endgroup$ – Graham Kemp Oct 29 '15 at 19:59
  • $\begingroup$ my mistake, i am not a native speaker :) $\endgroup$ – mkropkowski Oct 30 '15 at 12:45
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To combine the results we simply add them.

${1\over2}({1\over4}+{1\over4})+{1\over2}({4\over9}+{1\over9})={19\over36}$

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  • $\begingroup$ may I ask why did you do: 1/2(4/9+1/9). Not sure I got this part correctly. Thanks $\endgroup$ – adhg Nov 2 '17 at 1:37
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    $\begingroup$ @adhg The probability we choose the biased coin is $1\over 2$, the probability that it ends up with two heads is $({2\over 3})^2$ and similar for the two tails case $({1\over 3})^2$. $\endgroup$ – cr001 Nov 3 '17 at 2:15
  • $\begingroup$ thank you for the clarification! $\endgroup$ – adhg Nov 4 '17 at 2:23
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HINT: Let $S$ be the event of getting identical results, $F$ the event of choosing the fair coin, and $U$ the event of picking the unfair coin. Then

$$\Bbb P(S)=\Bbb P(S\mid E)\Bbb P(E)+\Bbb P(S\mid U)\Bbb P(U)\;.$$

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