A population consists of 25 men and 25 women. A simple random sample (draws at random without replacement) of 4 people is chosen. Find the chance that in the sample all the people are of the same gender?
Two events are dependent A and B.
Multiplication Rule P(A and B)= P(A)*P(B\A)=
if A and B are dependent then P(A and B)= P(A)P(B)=(4/25)(1)=?
Best Answer
If I understood correctly, without replacement means that after choosing someone, you cannot choose them again, so I will solve according to this.
Let's do this in steps.
We pick the first person and take them out of the room. They can be of any gender.
We pick the second person. They must be of the same gender of the first person. Since the first person is no longer in the room, there are now 49 people in the room, with 24 people of the first person's gender, such that the probability that the second person is of the same gender as the first is $ \frac{24}{49} $
We pick the third person, out of the room of now 48 people. There are now only 23 people of the same gender as the first person in the room, such that the probability that the third person is of the same gender as the first and second is $ \frac{23}{48}$.
Now, we pick the fourth person of a room of 47 people, which contains 22 people of the same gender as the first person, such that the probability that the fourth person will be of the same gender as the first person is $ \frac{22}{47}$.
Now, we want all of these events (meaning 1,2,3,4) to occur, and therefore, we need to multiply these probabilities, in order to get that the probability that all of the people are of the same gender is $ \frac{24\cdot 23 \cdot 22}{49\cdot 48 \cdot 47}$.
In general, using this algorithm, we can find a formula for the probability to choose k people of the same gender from a room of n men and n women.