# A group of twenty female and twenty male students are in a room. If eight of the #40# are selected at random and put in a row for a picture, what is the probability that the eight are of the same gender?

##### 2 Answers

#### Explanation:

So what I like to use is a tree diagram. Let me use a picture for better understanding

So what we do here is that we write in fractions. Let's say we have two balls in a cup, one red and one blue and then we get asked what is the procentage that we pick up the red two times. We can use a tree diagram for this in which we multiply 1/2 times 1/2. So in a tree diagram we take the numerator times the numerator and the denominator times the denomator to get the new fraction. See example in picture for better understanding.

So in this example there are an equal amount of genders ( boys and girls). So there is

Now make a tree diagram (it would be quite long to draw so just imagine) where we would take 20 times itself 8 times divided by 40 times itself 8 times to get the procentage. However this is only for one gender so you essentially have to dubble the numerator the get the procentage. Apply the following

However 1/256 is assuming there's only boys so dubble the numerator to get boys and girls.

Hope you understand and hope this helps =)

Approximately 0.3276%.

#### Explanation:

The probability will be

#"P"("8 same")#

#= ("num. groups of 8 males "+" num groups of 8 females")/"num. total possible groups"#

How many ways can we choose the group to be of exactly 8 males? This is choosing 8 men out of a pool of 20, which is

#""_20C_8=(20!)/(8!" "12!)="125,970"#

(The number of ways to choose 8 women out of a pool of 20 is the same.)

The number of ways to choose a random group of 8 people from a pool of 40 is

#""_40C_8 = (40!)/(8!" "32!)="76,904,685"#

Therefore, the probability of randomly choosing a group of 8 people and having them all be the same gender is

#"P"("8 same") = ("125,970 " + " 125,970")/"76,904,685"#

#color(white)("P"("8 same")) ~~ 0.003276#