A 12 member jury for a criminal case will be selected from a pool of 15 men and 15 women. What is the probability that the jury will have 6 men and 6 women?

Question

A 12 member jury for a criminal case will be selected from a pool of 15 men and 15 women.  What is the probability that the jury will have 6 men and 6 women?

Adrian Alexander · Accepted Answer

#~=.2896#

To find this, we need to know the number of ways the jury can be picked overall, and then the number of ways 6 men and 6 women can be on the jury. The fraction of the two will be the probability.

The total number of ways the jury can be picked is the combination of a pool of 30 people and choosing 12:

#C_(30,12)=(30!)/((12!)(30-12)!)=(30!)/((12!)(18!))#

and now let's evaluate this:

#(30xx29xx28xx27xx26xx25xx24xx23xx22xx21xx20xx19xxcancel(18!))/(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xxcancel(18!))#

#(color(red)(cancel(30))xx29xxcolor(blue)cancel(28)xxcolor(green)cancel(27)xxcolor(tan)cancel(26)^13xx25xxcolor(brown)cancel(24)xx23xxcolor(orange)cancel(22)^color(tan)cancel(2)xx21xxcolor(purple)cancel(20)^color(tan)cancel(2)xx19)/(color(brown)cancel(12)xxcolor(orange)cancel(11)xxcolor(purple)cancel(10)xxcolor(green)cancel(9)xxcolor(tan)cancel(8)xxcolor(blue)cancel(7)xxcolor(red)(cancel(6xx5))xxcolor(blue)cancel(4)xxcolor(green)cancel(3)xxcolor(brown)cancel(2))#

#29xx13xx25xx23xx21xx19=86,493,225#

Ok - we know the number of ways the jury can be picked. Now how many ways can the jury consist of 6 men and 6 women?

For the men, there are 15 in the pool and we're picking 6:

#C_(15,6)=(15!)/((6!)(15-6)!)=(15!)/((6!)(9!))#

and now let's evaluate that:

#(15xx14xx13xx12xx11xx10xxcancel(9!))/(6xx5xx4xx3xx2xxcancel(9!))#

#(color(red)cancel(15)xxcolor(green)cancel(14)^7xx13xxcolor(blue)cancel(12)xx11xxcolor(green)cancel(10)^5)/(color(blue)cancel(6)xxcolor(red)cancel(5)xxcolor(green)cancel(4)xxcolor(red)cancel(3)xxcolor(blue)cancel(2))#

#7xx13xx11xx5=5005#

The same number of choices of women are available, so we multiply the number of men's choices and women's choices:

#5005xx5005=25,050,025#

And now we can find the probability:

#(25,050,025)/(86,493,225)~=.2896#

Audrey Allen · Answer

undefined The probability that the jury will have 6 men and 6 women is 0.1966, or approximately 19.66%.