A) \[{{H}_{2}}\] and \[{{A}_{6}}\] are independent
B) \[{{H}_{7}}\] and \[{{H}_{2}}\]are independent
C) \[{{H}_{4}}\] and \[{{A}_{3}}\] are independent
D) \[{{H}_{2}}\] and \[{{A}_{5}}\] are independent
Correct Answer: D
Solution :
[d] According to the question, \[P({{H}_{i}})=\frac{1}{2}\] and \[P({{A}_{m}})=\frac{^{10}{{C}_{m}}}{{{2}^{10}}}.\] Also, \[P({{H}_{i}}\cap {{A}_{m}})=\frac{^{9}{{C}_{m-1}}}{{{2}^{10}}}\] Now, \[{{H}_{i}}\] and \[{{A}_{m}}\] to be independent events, if \[\frac{^{9}{{C}_{m-1}}}{{{2}^{10}}}=\frac{1}{2}\times \frac{^{10}{{C}_{m}}}{{{2}^{10}}}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\frac{^{10}{{C}_{m}}}{^{9}{{C}_{m-1}}}=2\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\frac{\frac{10!}{m!(10-m)!}}{\frac{9!}{(10-m)!(m-1)!}}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\frac{10}{m}=2\Rightarrow m=5\]
You need to login to perform this action.
You will be redirected in
3 sec
