If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men?A. 24/91B. 45/91C. 2/3D. 67/91E. 84/91

Answer:D. [tex]\frac{67}{91}[/tex]Step-by-step explanation: Given,Total number of jurors = 15,Out of which 2/3 are men and 1/3 are women,So, total men = [tex]\frac{2}{3}\times 15[/tex] = 10,And, total women = 15 - 10 = 5,The people have to select = 12,So, the total ways of selecting 12 people = [tex]^{15}C_{12}=\frac{15!}{12!\times 3!}=455[/tex]Now, 2/3 of 12 = [tex]\frac{2}{3}\times 12[/tex]  = 2 × 4 = 8,Thus, there would be at least 8 men.Now, the possible ways of selecting at least 8 men = 8 men 4 women + 9 men 3 women + 10 men 2 women[tex]=^{10}C_8\times ^5C_4 + ^{10}C_9\times ^5C_3+^{10}C_{10}\times ^5C_2[/tex][tex]=\frac{10!}{8!\times 2!}\times \frac{5!}{4!\times 1!}+\frac{10!}{9!\times 1!}\times \frac{5!}{3!\times 2!}+\frac{10!}{10!\times 0!}\times \frac{5!}{2!\times 3!}[/tex]= 225 + 100 + 10= 335Hence, the probability that the jury will comprise at least 2/3 men [tex]=\frac{\text{the possible ways of selecting at least 8 men}}{\text{total ways of selecting 12 people}}[/tex][tex]=\frac{335}{455}[/tex][tex]=\frac{67}{91}[/tex]OPTION D is correct.