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Four fair dice D1,D2,D3,D4 each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously . The probability that D4 shows a number appearing on one D1 ,D2 ,D3 .

I could not able to start . I am confused .

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    $\begingroup$ Hint: the probability that D4 shows 1 and D1,D2,D3 do not show 1 is ...? $\endgroup$ – Yurii Savchuk Apr 25 '17 at 15:36
  • $\begingroup$ @YuriiSavchuk then the answer should 125/216 . But the answer given as 91/216 $\endgroup$ – search Apr 25 '17 at 15:48
  • $\begingroup$ Is it that D4 shows a number that’s on any of the other dice or on exactly one? $\endgroup$ – amd Apr 25 '17 at 18:32
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Note that the probability of the event $\{D_4$ is equal to one of the other three dice rolls} depends only on the number of distinct rolls among $D_1,D_2,D_3$; call the number of such distinct rolls $N$. Then we can compute the probability of $N=1,2,3.$ We also know the probability of {$D_4$ being equal to one of the other three dice rolls, given the value of $N$}. Use the law of total probability to get the result (https://en.wikipedia.org/wiki/Law_of_total_probability).

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  • $\begingroup$ it is rather a hint $\endgroup$ – Yurii Savchuk Apr 26 '17 at 15:08
  • $\begingroup$ @YuriiSavchuk More of an outline - but we are splitting hairs. $\endgroup$ – manofbear Apr 26 '17 at 20:05

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