# Probability of $D_4$

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 .

• Hint: the probability that D4 shows 1 and D1,D2,D3 do not show 1 is ...? – Yurii Savchuk Apr 25 '17 at 15:36
• @YuriiSavchuk then the answer should 125/216 . But the answer given as 91/216 – search Apr 25 '17 at 15:48
• Is it that D4 shows a number that’s on any of the other dice or on exactly one? – amd Apr 25 '17 at 18:32

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).

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