Question

four fair dice `D_1, D_2,D_3 and D_4` each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously. The probability that `D_4` shows a number appearing on one of `D_1,D_2,D_3` is A. `(91)/(216)` B. `(108)/(216)` C. `(125)/(216)` D. `(127)/(216)` Select the correct answer from above options

Answer 1

Correct Answer - A As one of the dice shows a number appearing on one of `P_1, P_2 and P_3` . Thus, three cases arise. (i) All show same number. (ii) Number appearing on `D_4` oppears on any one of `D_1,D_2 and D_3` (iii) Number appearing on `D_4` appears on any two of `D_1,D_2 and D_3` Sample space `=6xx6xx6xx6=6^4` favorite events =Case I or Case II or Case III CaseI First we should select one number for `D_4` winch appears on all i.e `^6C_1xx1` Case II For `D_4` there are `^6C_1` ways. Now, it appears on any one of `D_1,D_2 and D_3 i.e ""^3C_1xx1` For other two there are `5xx5` ways `rArr ^6C_1 xx^3C_1xx1xx5xx5` For `D_4` there are `^6C_1` ways now it apperes on any two of `D_1,D_2 and D_3` `rArr ^3C_2xx1^2` For another one there are 5 ways `rArr ""^6C_1xx""^3C_2xx1^2xx5` Thus ,Probability `=(""^6C_1+""^6C_1xx^3C_1xx5^2+""^6C_1xx""^3C_2xx5)/(6^4)` `=(6(1+75+15))/6^4` `=91/216`